HAL Id: jpa-00247545
https://hal.archives-ouvertes.fr/jpa-00247545
Submitted on 1 Jan 1991
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Adsorption of star polymers
A. Halperin, J. Joanny
To cite this version:
A. Halperin, J. Joanny. Adsorption of star polymers. Journal de Physique II, EDP Sciences, 1991, 1 (6), pp.623-636. �10.1051/jp2:1991194�. �jpa-00247545�
J Phjs II1 (1991) 623-636 JUIN 1991, PAGE 623
Classification Physics Abstracts
61 25H 68.45
Adsorption of star polymers
A Halpenn (I) and J. F. Joanny f)
(Ii Max-Planck-Institut fur Polymerforschung Postfach 3148, W-6500 Maim, F R.G.
f) Institut Charles Sadron, 6, rue Boussingault, F-67083 Strasbourg Cedex, France (Received 25 January 1991, accepted 6 March J991)
Rksumk. Nous ktudions l'adsorption de polymdres en dtoile sur une surface sohde en utilisant une approche de lois d'kchelles basde sur le moddle de blobs de Daoud et Cotton. Pour une dtoile formbe de fbras contenant chacun N monomdres, nous distinguons trois rdglmes suivant la valeur de l'dnergie d'adsorption d'un monomdre sur la surface &kT I) L'adsorption forte caractdriske
par une adsorption complete de tous les bras 5e produit lorsque
~ ~f/N)~'~ 2) Une structure en
« sombrero » avec f~~~ bras adsorbks et f f~ds bras hbres est obtenue si f~'~°fN~'~ < ~f/N)~'~
3) Les diodes faiblement adsorbdes gardent une structure trds similaire I celle des ktoiles hbres
en solution Ce rdgime exlste si < f~/~°/N~'~ La structure correspondant aux dtoiles faiblement adsorbdes peut aussi exlster comme un etat mdtastable si
~ f~/~/N~/~
Abstract. The adsorption of star polymers on a flat solid surface is analyzed by means of scaling arguments based on the Daoud-Cotton blob model For the adsorption of a single star, consisting
of f arms compnsing each N monomers, we distinguish three regimes determined by the
adsorption energy of a monomer at the surface, &kT I) Strong adsorption charactenzed by the full adsorption of all arms occurs for
~ f/N)~/~ 2) A « Sombrero like structure compnsing f~ds fully adsorbed arms and f- f~~~ free arms 15 obtained for ~f/N)~/~~&
~ f~/~°/N~/~
3) Weakly adsorbed stars retain, essentially, the structure of a free star Thls regime occurs for
&<f~/~%N~/~ The weakly adsorbed structure may also exist as a metastable state if
~ f9/5jfiT3/5
t. Introducdon.
The study of polymer adsorption is mostly concerned with linear chains. Much le55 attention has been given to the adsorption of branched polymers in general and of star polymers in
particular Yet, this problem is of interest for practical as well a5 fundamental reasons The
adsorption behavior of star polymers and linear chains differ in an important way The deformation of linear chains upon adsorption is due, solely to the interaction with the surface In marked contrast, the coils compnsing the arms of a free star are already deformed because of the repulsive arm-arm interactions [1, 2, 3, 15] Thus, the adsorption of star polymers
involves an interplay of two types of deformations, due respectively to molecular architecture and to surface interactions. This balance is of interest because of its impact on the interfacial behavior of stars. The elucidation of star adsorption is also of crucial importance to the
understanding of the adsorption of star like micelles [4, 5]. These are micelles endowed with
624 JOURNAL DE PHYSIQUE II M 6
extended coronas which are structurally similar to star polymers. The adsorption of such mlcelles, when the corona is attracted by the surface, is a possible mechanism for the formation of monolayers of adsorbed block copolymers In tum, these are of practical importance to a vanety of applications colloidal stabilization, compatibilizatlon etc. [6, 7].
In the following we consider the adsorption of star polymers onto a flat surface immersed in
a good solvent of low molecular weight The stars are assumed to be monodispersed,
compnsing farms such that each arm consists of N monomers Their interaction with the surface is considered as a function of the adsorption energy of a monomer at the surface,
kT3 We also assume that the core of the star has a small volume and plays no specific role
in the adsorption. In particular, the case of a star anchored to the surface because of strong
attraction between the core and the surface is not studied.
The next section reviews the relevant results conceming the behavior of free and confined
stars. It surwnadzes the Daoud-Cotton model of free stars and its adaptation to stars confined
to slits. Th1s is followed by discussion of single star adsorption. Three regimes occur as 3 is vaned. Strong adsorption in which all arms are fully adsorbed, an intermediate regime where only some of the arms adsorb and, finally, weak adsorption where only some of the arms are
partly adsorbed. We then present some conjectures on the adsorption isotherm and a few
concluding remarks. The two dimensional case, of relevance to simulations, is summarized in the appendix.
2. Free and confined stars.
Our description of star polymers is based on the Daoud-Cotton model ill. The free star is
depicted as a spherical region containing close packed blobs arranged m concentnc, sphencal
shells The blobs in a given shell are of constant size, f (r), set by the radius of the shell. Each of the star's farms is assumed to contnbute a single blob to any given shell This is a crucial
requirement, allowing for the stretched configurations adopted by densely grafted chains. The volume of a given shell, r~f, is thus equal to ff~, the volume of the constituent blobs.
Accordingly the correlation length of a free star, fi(r), is given by
fi(r) r/f~'~ (2.1)
In the Flory approxJmation, the fractal dimension of a linear chain in a good solvent is
di = 5/3 and the number of monomers in a blob of size f is (fla)~'~ The local monomer volume fraction is then
4 (r)
»
f~°(air)+~. (2.2)
Monomer conservation determines the radius of the free star, llo
llo m f "~ N~'~a (2.3)
The kT per blob prescnption yields the star free energy F (actually, its excess free energy with respect to f free linear chaJns consisting each of N monomers)
F/kT lR
m 11 r~ dr
m f ~'~ln (R + R~~~~)/R~~~ m f~'~ (2 4)
Rwm
where Jl~~~ is the radius of the inner, ordered, region compnsing the central multi-functional
monomer
M 6 ADSORPTION OF STAR POLYMERS 625
A s1rnllar approach may be used to stars confined to a slit of width D between two non- adsorbing walls [8] Again, the star is pictured as a layered conglomerate of blobs such that each arm contributes a single blob to any given layer However, the confinement modifies the geometry of the layers and the structure of the blobs. As a result we may distinguish between three regions
(I) An inner region reta1nlng the structure of a free star, occurs for r
~ D In this region f is given by f~
~
r/f"I
(ii) For r ~D there occurs an intermediate region charactenzed by loss of the overall
sphencal symmetry while retaining blobs' three dimensional structure. In particular, the blobs may be considered as spheres of volume f~, but the shells are now cylindncal. As a result we have rDf
m ff ~ or
f, (rD/f)"~. (2.s)
Thls region extends up to r
m fD where f, m D and the confinement begins to afsect the blobs structure.
(iii) An extenor, two dimensional region extends beyond r
m fD. It is charactenzed by cylindncal rather than sphencal symmetry. In this region the blobs are two dimensional, of
volume if D, where iii is the in-plane correlation length. The Daoud-Cotton construction
yields rf
ii
D
m ff D or
ii m r/f (2.6)
Each two dimensional iii blob compnses a stnng of gii three dimensional blobs of size D set by
the slit width. This string exhibits the behavior of a two dimensional self-avoiding random
walk i-e-, ii mgi'~D. Accordingly, the total number of monomers in the iii blob is
(f
ii
ID)~/~ (Dla)~'~ In the limit of large N, when the two dimensional region dominates the star behavior, the star in-plane radius, Rjj, is given by
~ ~ij~ ~~/~~~j~ ij~ ~~ ~~
3. The strong adsorpdon limit.
A strongly adsorbed star forms a flat pancake of thickness D (Fig. I). All farms are fully
adsorbed and the star center is confined witlun a distance D from the surface. ThJs state is
reminiscent of confinement to a slit of width D However, in this case D is not imposed by an
extemal geometric constraint Rather, it is set by the adsorption energy per adsorbed
monomer, kT3 The free energy of an adsorbed star in this regime consists of two terms.
Fig. I Structure of an adsorbed star polymer in tile strong adsorption regime.
626 JOURNAL DE PHYSIQUE II N 6
The first, F~, allows for the deformation of the star and is identical to the confinement free energy of a star in a slit The second term, F~, accounts for the adsorption free energy Clearly, F~ is proportional to the number of monomers in contact with the surface The
calculation of this number does not follow the intuitive prescnption used in early theories
concerned with the adsorption of linear chains These assumed that the monomers are
uniformly distributed throughout the layer. Consequently, the total number of monomers per unit area within a layer of tluckness z (a « z « D ) scales as z/D and the fraction of monomers at the surfaces scales as a/D In time it was recognized [9, 10] that this approach must be
modified to allow for the fact that each monomer at the surface drags its neighbors with it. As
a result the monomer volume fraction within a layer of thickness z is not constant but decays algebraically as (z/D)~ with an exponent m w 1/3. This picture may be reformulated in terms of adsorption blobs [I I]. These are isotropic chain segments of size D exhibiting the behavior of a free coil Thus, in a good solvent each blob consists of g
m (Dla)~'~ of monomers. In each
adsorption blob, only g~ of the g monomers are in contact with the surface where
#
= I + (m I )/dim 3/5 is the crossover exponent. The adsorbed state of a single linear chain may be fully descnbed in terms of such blobs It is pictured as a self avoiding, two dimensional stnng of D blobs As we shall see the situation in our case is more complicated
because of the star topology and the resulting interactions between the arms. For brevity we first analyze the confinement free energy.
The confinement free energy is proportional to the overall number of blobs in the three
« stellar regions described in the previous section. The number of blobs in a given region is found by integrating the appropriate blob density In the extenor region one must account for two types of blobs D blobs arising from the confinement of each arm and two dimensional iii blobs reflecting the interaction between the farms which form a two dimensional star
Altogether we find
D fD R
Fc/kT ~
~ ~f ~~~ dr + j~ f~ rD dr +
~~ Ii + (fl /l~)~~~](~I l~ ~l~ d~
Fc/kT ~
f~Tl (D/r~~e f~~~
+ f~ in ~r~~'~(Q/D)~'~ N ~"j + f (a/D)~'~ N
~~
Since the last two terms are clearly dominant we will use
F~~~~ » fN (aiD)5/3 + f2 (3.2)
where the loganthmic factor multiplying f~ has been approximated by a constant.
Only three dimensional blobs in contact with the surface are relevant to the adsorption free energy, F~ Each blob, compnsing g monomers, contains g~ monomers in contact with the surface The corresponding contribution to F~ is thus g~ kT3. Accordingly, the adsorption
free energy due to the two interior regions is obtained by summing the contributions due to all surface grazing adsorption blobs
fD R
Faj8kT ~ (fi/Q) f> ~~ dr + (D/Q) (f
iID)~~~iii ~ r dr m 3 fN (Q/D)~~~ (3-3)
D fD
Altogether, the stars free energy, F
= F~ + F~ is given by F/kT m
f~ + fN (a/D)~'~ 3fN (a/D)~'~ (3.4)
Minimization with respect to D yields
D=3-~a, (3.5)
bt 6 ADSORPTION OF STAR POLYMERS 627
a result familiar from the study of linear adsorbed chaJns [12]. This reflects the dominance of
the extemal region contribution to F which is due pnmanly to single ann behavJor By
combining the last two results we obtain the total free energy df a strongly adsorbed star
F/kT m f~ fN3 ~'~ (3.6)
The strong adsorption limit is dominated by the contributions of the extemal region Clearly
this description is only meaningful if the extemal region of the star indeed exJsts I-e- R » fD Thls condition defines cross over values of D and 3, D~
= Rj /f and 3~
= a/D~ given by
D~ m
(Nif)~'~ a
,
3
~ m
~fiN)~'~ (3 7)
For 3
~ 3~ or D
» D
~
the strong adsorption limit can no longer be obtained and the absorbed star adopts the « Sombrero » configuration discussed in the next section.
R o
o
q i1
Fig 2 Structure of an adsorbed star in the « Sombrero » regrme
4. The Sombrero structure.
When 3 falls below 3~ the flat pancake structure, charactenstic of the strong adsorption regime, is replaced by a «Sombrero» configuration (Fig. 2). In this regime only f~~~ ~ f arms are adsorbed. These form a confined star structure of the type considered m the
previous section i e., a flat pancake, of thickness D 3 a, incorporating the star center.
Tile remaining fi~~
= f f~~~ arms are free These adopt a free star configuration but occupy
a hemisphere rather than a sphere The free arms are stretched according to the Daoud- Cotton model As we shall see the radius of the adsorbed layer is much larger than that of the
hemisphere hence the name Sombrero ». the free energy of this structure contains two terms, Fi~~ and F~~~, accounting respectively for the contributions of the free and adsorbed
arms. Fi~~ is given by (34) with fi~~ replacing F F~~~ is obtained from (3.8) once
f~~~ takes the place of f. Accordingly, F is given by F/kT
"
fill + f]ds fads N3 ~~~ (4.1)
628 JOURNAL DE PHYSIQUE II M 6
The minimization with respect to f~~~, which amounts to equating the chemical potential of the free and the adsorbed arms, leads to
~f fads)~~~ fads + N3 ~~~
"
° (4 2)
For f~'~«f~~~ s f the first term is negligible and f~~~ is given by
fads ~ N& ~~~ (4.3)
This result is also obtained if we use the cntenon D
=
D
~
to set the number at adsorbed arms.
In other words the adsorbed layer thickness is maintained at D by desorption of arms and D m (N/f~~~) a~'~ leads to f~~~ m N/(Dla)~'~
m N3 ~'~
The honzontal radius of the star is obtained from equation (2 7)
Rji m N3 ~'~ (4.4)
or Rjj m f~~~D. J§j is indeed larger than the radius of the hemispheres, Rm f/($ N~'~a, provided D
= 3 a
~ f
~ where f~
~
N ~'~fj~('~° is the size of the largest blob of the Daoud- Cotton structure in the hemisphere Equation (4.3) also determines the free energy of the
« Sombrero structure
F/kT
m
N~3 ~°'~
+ f~'~
m Rj)/D~ + f~'~. (4.5)
This free energy is positive if 3 ~ f~'~°N~~'~ and becomes equal to the free energy of a free star confined in a half space at small values of 3. The Sombrero structure exists only at intermediate values of the adsorption energy 3. 3 must be smaller than 3~ but we also required f~~~ » f~'~I e., 3
»
N~~'~f~'~° In terms of the pancake tluckness D, the sombrero exJsts if D~ ~ D ~ f
~. If D
» f
~ the star is only very weakly adsorbed, a situation we do not consider further
2R~
D t
Fig 3 Structure of an adsorbed star in tile weak adsorption limit.
bt 6 ADSORPTION OF STAR POLYMERS 629
5. Weakly adsorbed stars.
For sufficiently small adsorption energy, 3, the adsorbed star is only weakly deformed only
some of the arms are partly adsorbed. In this limit it is helpful to view the star as a droplet
of a semldilute solution The adsorbed monomers are not expected, thus, to develop a two dimensional Daoud-Cotton structure. Rather, they form a uniform semldilute surface layer
The adsorbed monomers originate in a small sphencal cap charactenzed by an apex angle of 2 @o « ar/2 (Fig. 3). The boundaries of the cap and the surface layer are assumed to coincide
i e, the layer in-plane radius, L, is given by L mR sin @o mR@o. The number of partly
adsorbed arms, f~~~, is thus f~~~ m f@) The remaJnder of the star is unperturbed. A full charactenzation of this state is possible if one assumes, as in the case of a sernldilute solution, that the adsorbed layer consists of close packed D blobs. The area of the adsorbed layer, L~
m
R~ @),
is than equal to the total area of the D blobs, (n/g) D~, n being the number of adsorbed monomers This requirement determines
@o and allows for a complete charactenza- tion of this adsorption regime In order to carry out this program it is necessary to find the
number of adsorbed monomers n, or equivalently, the number of monomers in the sphencal
cap For eon ar/2, the regime of interest, n is given by
nm fN@(. (5.1)
Accordingly
f- 3/10fi~1/10( / J~)1/6 f- 3/10~~f1/10 ~ l/6 (~~)
0 ~ ~ "
The number of adsorbed monomers is thus
~ f 1/5~~f7f5~2f3 (~ ~)
L is given by
~ ~ f- lf10fi~7f10flf6 (~ ~)
0 0
and the number of partly adsorbed arms is
fads ~ f@]
~ f~~~N ~~~3 ~~ (s.s)
Each of these arms contributes N~~~ monomers to the adsorbed layer
~~f ~/ f f 3/5~~f6/5~l/3 (~ ~)
ads ads
Within this picture the free energy of a weakly adsorbed star is
F/kT m
f~'~ (n/g g~ 3
m f ~'~ f ~'~ N ?'~ 3 ?'~ (5 7)
The first term, the free energy of a free star, approximates the free energy of the unperturbed region of the adsorbed star. It is supplemented by the adsorption free energy of the n
monomers which may be written as kTL~/D~.
One may gain some insight by considering a different route to these results It calls for minimization of an appropriate free energy To account for the contributions of the adsorbed layer we view it as a close packed array of two dimensional blobs of size iii Each iii blob compnses a self avoiding, two dimensional string of D blobs. The monomer area
fraction within the adsorbed layer is r
m
na~/L~mf~'~N~ ~'~@). This is also the
monomer
