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Theory of the adsorption of a macromolecule on a plane surface
Polymer Science U.S.S.R. Vol. 29, No. 10, pp. 2430-2437, 1987 Printed in Poland
0032-3950/87 $10.00+.00 © 1988 Pergamon Press pie
THEORY OF THE ADSORPTION OF A MACROMOLECULE ON A PLANE SURFACE* M. L. POLYAKOV Physicotechnical I,-~stitute for Low Temperatures, Ukr.S.S.R. Academy of Sciences
(Received 12 May 1986) The adsorption of rigid macromolecules on a plain surface is analysed, and the rigidity of a macromolecule is described within the framework of a three-dimensional lattice model with a correlation belween ~he directions of adjacent monomers. The equilibrium characteristics of a macro:nolecule arc calculated, a~:d an ex!;ression for the free energy is derived. An increase in chain rigidity imparts a cooperative charader 1o the bonding of its monomers to the surface. Large fluctuations in the macroscopic values of the lengths of the adsorbed fragments and rings are a feature of the system. SIGNIFICANT success in studying the equilibrium properties o f absolutely flexible macromolecules interacting with a surface has been attained recently. A c c u r a t e results for a lattice model o f an ideal chain with an equilibrium distribution o f adjacent m o n o m e r s [1 ] have beert obtained. Qualitative interpretation o f the relation between the free energy o f a macromolecule and the adsorption energy [2] provides a means o f establishing the main laws, and also of allowing for the effect of th~ excluded vo!a~ne. A n o t h e r reasonable generalization o f the simplest model consists in allowing for the rigidity (this applies firstly to biomacromolecuies), i.e. the correlation between the directions o f the adjacent m o n o m e r s along the chain [3]. Nevertheless, a knowledge not only of the m e a n characteristics but also their fluctuations is necessary for a more complete understanding of the properties and special features of the system. In most works touching on the adsorption o f macromolecules fluctuations are not cortsidered. On the other hand, calculation o f the fluctuations of the m e a n values is o f interst for at least two reasons. Firstly, attraction of a macromolecule to a surface with a fairly high energy of interaction can result in the formation of a specific state. This state o f the macromolecule is, in a certain sense, intermediate between the state o f a coil and a globule, which differ in their fluctuation conditions [4]. Secondly, for c o m p a r i s o n o f the conclusions o f theory with the results of the corresponding experiments, for example by adsorptiort c h r o m a t o g r a p h y , only values with small fluctuations can be used. In this w o r k the special features of tb.e adsorption of a rigid m a c r o m o l e c u l e on a plane surface are studied, with. the object of analyzing the relation between the mean characteristics of the macromolecule a n d their fluctuations as a function o f the chain rigidity. The use of an accurately determined three-dimensional lattice molecule with. * Vysokomol. soyed. A29: No. 10, 2212-2218, 1987. 2430
Theory of the adsorption of macromolecule on a plane surface
2431
correlation between the directions of the adjacent monomers provides a means not only of calculating these values accurately, but also of expressing them in terms of a macroscopic parameter, i.e. the presistent chain length. Model. Statistical sum o f a macromolecule. It is suggested that all tire monomer units in the macromolecule have the same length a. Tire stiffness of the chain is manifested in the lact that configurations with different angles between adiacent monomers have different energies. The discussion will be restricted to the simplest case, when the configuratimml energy is equal to N-1
u= i=1
where tr~ is a unit vector directed along the i-th monomer, and [ is a parameter determiuing the excess chain stiffness. In a three-dimensional lattice model trz can have only six values:, i.e. ~rz--=+_n,., _+ny, ___n:, where n,, ny, and n: are the unit vectors of the coordinate axes. This model coincides with the model assumed by Birshtein cta/. I3]The quantity l determines the important macroscopic ch,.tracteri.~tic of the macromolecular coil, and the presistent length. The persistent length of the macromolccule 1~, is equal to the length along the chain for which a correlation betx~een the directions of the monomers disappears (is decreased by a factor of e). When I>'.T, i.e. for a rigid chain Ip=cca/4 , c~=exp(l/T) (2) The coordinate axes x, and y lying on the adsorbing surface will be selected. The interaction between the monomer units of the macromolecule with this surface can then be described by means of an external field V(z): I/(=)= =to when z < 0 (this condition corresponds to an absolutely impermeable face, since the proba~'lity of finding a unit at a point with a coordinate z is proportional to exp ( - V(z)/T)), K (z)= - l/ (V>0) at - = 0 , and V ( z ) = 0 at z > 0 , where the coordianates of the plane surface ==0, and k" is the sorption energy (an energetically more favourable unit is fouud on the adsorbing surface). Grosberg [4] used a shmlar external field for an:-t!ysis oi adsorption. Using the method proposed by Kosevich el al. [5] it is possible to find a clear expression tbr the statistical sum of the chain from N monomer tmlts with fixed z and z' coordinates and ~ and 6' directions of the initial a n d final units, ZN
,
~r)~j(z'-ag'n., -6')Bj,
=7N-t~e-ae°v-l)~j(z,
(3)
J where V= 1 + 4/'c(+ 1/e z,
E Eexp{v(a,,,~/r+ V(am-a~n.~/T} ~,~
B f ~=
m=O
a
(am, ~) ~ ( a m - a n : g , - g ) ; 2j and ~/~.(z, or) are the intrinsic values and corresponding
intrinsic functions of the equation q/j.(z, o') exp { - 2j + V ( z ) / T } = ~ a (6s) ~ j ( - - aon~, s) s
g (as) = ego e x p ( I 6 s / T ) ,
# o ~ = ez + 4~ + 1
(4)
M.L. POLYAKOV
2432
Accordingly, the problem ol determining the statisticalsum is reduced to the proplem of solving eqn. (4). ChemicaIpotential and size of effectivelayer. In order to analyse the equilibrium properties of a rigidmacromolccule it is necessary to solve the system of cquations (4) for the values ~ (z, ~), where ~ = &nx, +ny, ±nz. The minimum value of 2, which corresponds to the function ~o (z, ~), is of special interest,this decreasing exponentially on moving away from the adsorbing surface ~/o(z, o)- A (~) exp (- kz) at z I>a. The values of 2o and k can be found by analyzing the behaviour of ~o (z, a) at the boundary from eqn. (4) when z=O, a. Using ~o(z, ~) at z<0, it is not difficultto obtain such a system of equations I (.4-I) [.4-(~2- I)a0] [A-(~- I)29o]
2(c°shka-1)=Cdgo
A[A-(o~-l)(a+3)go]
Aexp(-ka)-(~2-1)go At At-(~+l)Zgo A - (a2 _ 1) g0 exp ( - ka)- 9o At + (3a + 1) ( ~ - 1) 0o
(5)
where A = exp (7- 2o), t = exp ( - V/T). It is not difficult to be convinced that the values of 20 and K determine the chemical potential of a monomer unit and the size of the effective layer respectively, in which almost the whole of the chain is concentrated. In fact, it follows from eqn. (3) that on approaching the dominating state of the chain [2] the free energy of the macromolecule, as calculated from the energy of the macromolecular coil F o = - N T I n 7, for N>>I is equal to
AF = F - F 0 = #N = T2o N ,
(6)
where/t is the chemical potential of the unit. The equilibrium density of the monomer units of the macromolecule irt this case is
p(z, ~)---NBotYo(z-aon,, -a)~o(Z,
•
(V(z)+V(z-aern,)]
~)exp t
~
j~
(7)
It follows from eqn. (7) that the contribution from the adsorbing surface is p (z, ~) ,-,exp ( - 2 k z ) . Since at a distance hcf h,f = (2k)- 1
(8)
the density of the units is decreased by a factor of e, it is natural to consider it equal to the size of the effective layer, in whichalmost the whole of the macromolecule is concentrated. A similar quantity is introduced by De Gennes [2] for an absolutely flexible chain. Distribution functions for an adsorbed section and a ring. When adsorbing surfaces are present the maeromoleeule consists of alternating adsorbed (contacting the surface) sections and rings joining adjacent adsorbed sections. The attraction of the monomer units to the surface results in a situation where at zero temperature the formation of an adsorbed monolayer is most favoured. On raising the temperature macromolecule sections are formed which do not contact (that is do not interact) with the surface, i.e.
Theory of the adsorption of' macromolecute on a plane surface
2433
the ring. This is because the entropy of the adsorbed section is less than that of a ring of the same length. To determine the equilibrium characteristics of the adsorbed section and ring it is convenient to use the method of a large canonical assembly [6]. The large statistical sum of the macromolecule can be represented in the form Z (/0 = {[ZI(/~)] -~ - Z"(/z)} - i,
,.
(9)
where Z l is the large statistical sum of the adsorbed section Z ' = go exp {(i, + V)/T} 1 + ( a - 1) (3a + 1) go exp {(#+V)/T_! 1 - ( a + 1)Zgo exp {(/2+ V)/T} '
(1 o)
and Z ~Jis the large statistical sum of the ring zU= ~ W(n)exp {fin/T},
(ll)
n= 1
where W (n) is the probability that the chain consists of n monomeric units, beginning in the plane z = a , and (n + 1)junctions make contact with the adsorbing surface only once. It follows from eqn. (11) that as a result of allowing for the formation of rings in the macromolecule long range entropy makes its appearance. Such long range action is not restricted to the ultimate n, and in the case of macromolecules of infinite length results in a phase transition [7]. The quantity W(n) determines the convergence of the series (11) and its derivatives with respect to /2. In its turn, the smoothness of the thermodynamic characteristics of the system, and consequently the existence and form of the phase transitions depend on the convergence of the given series. Unfortunately, direct calculation of the function W (n) involves formidable mathematical difficulties and has not been carried out even for a network model of an absolutely flexible polymer. A simpler method of studying the equilibrium characteristics of a ring consists in calculating the sum (11), i.e. the generating function for W(n), and in order to do this certain properties of the system under consideration are used. Firstly, the distribution function Z H is not clearly dependent on the sorption energy V. Secondly, the chemical potential of a unit of an endless chain N = oo can be found from the equation Z ~- Z u = 1 [6]. The given chemical potential will naturally be equal to the previously found chemical potential ,u, as given by eqn. (6). In order to determine Z t~, it is thus necessary to find, by means of eqn. (5), the values of V as a function of a and/2, and then to insert them in.{ZI} -1. After somewhat tedious but not complex transformations, the final result is ZH = exp ( - ka) - (a 2 - 1) go exp (p/T) (12) 1 - (a 2 -- 1) go exp ( - ka + p/T) where the value k is a function of/2 (system of eqn. (5)). Knowing the distribution function for the adsorbed section and the ring it is possible
2434
M.L. POLYAKOV
to find their mean length, and also the fluctuations of these values
li=ar-~lnZi(v), wi=ar
lnZ'(~)
I,,
(13)
where i= I, II for the adsorbed section and the ring respectively. In the remainder of this paper the discussion will be restricted to analysis and discussion of the following quantities as a function of the rigidity and sorption energy: the chemical potential p, the size of the effective layer h~f, the mean lengths of the adsorbed section Ii and the ring l~T,the relative fluctuations of these mean values w~ and ,wH, and also the number of boundaries between the adsorbed sections and. the rings
Nb = N { r ff---~[Z'(l~)+ Z"(p)]} - I
(14)
Analysis of results. For a thermodynamic description of the properties of the macromolecule it is necessary to solve the system of equations (5), i.e. to find the minimum inherent value of 20 and the parameter k, on which tire density of the monomers depends. An important feature of these equations will first be singled out, i.e. at the critical sorption energy (or critical temperature) Vc= - r In
cz2+ 2 e + 2 + ~ ( ~ 2 +4~+20) ~2 (c~/+4c~+ 1)
(15)
where 20 and k vanish. Moreover the energy of the macromolecule is equal to the energy of a macromolecular coil in free space. It is noted that under the given critical conditions of sorption the approximation of the doininating state of the chain is true only for N ~ o o . When V> Vc a state of the macromolecule in which the number of contacts with the surface N,a is proportional to N is more favourable, i.e. Nad=ON =
- N - dd# ~,
(I 6)
where 0 is the degree of sorption. When however V< V~ the chain is located mainly far away from the surface and (Nao/n)-~Owhen N ~ ~ . Because of the complexity of the system of eqn. (5) it is necessary to find a relation for the equilibrium characteristics of the molecule only in certain limiting cases. Table 1 gives a relation for these characteristics as a function of the rigidity and sorption energy when the chain rigidity is high, a>>l, and V~=2T/a. The values of the equilibrium characteristics of the macromolecule given in Table 1 bring out an interesting feature of the system under consideration. The fluctuations of certain macroscopic quantities, e.g. the number of adsorbed monomers Nad (Nao>>l), and the interface between an adsorbed section and a ring N~. (N~,>>I) are small and w,,~1/x/N, where N is equal to Nan or N~, respectively. Thus the behaviour of these quantities is the same as in most macroscopic systems [8]. On the other hand, although the adsorbed section and the ring are also macroscopic and l~>>a, /~>>a, their fluctuations are large even far away from the critical conditions of phase transition (15). In
<<
V-
1
~
3
V
a<
--<<--<< 1 T
1
a
lp<< ha~< - -
~<< ----<< ct Z
1
a
h~f~ f e
T
V - Vc
~t
1 --
Regions o f c h a n g e in a d s o r p t i o n energy
a2
i2h~f
alp
a
412
a
2ha
2
81~
2 2 a her
-- 12h2f "4 4lp her
alp
__
It T
I
'
~*
2 ( v - ~)
aT
V--i o)j
alp T
41p( V - V~)
a2T
hcf
8
ah 2f
14 "P
a
tl
WI
4ha
6her
6h~f
Ill
1
)4111
R E S U L T S OF C A L C U L A T I N G THE EQUILIBRIUM CHARACTERISTICS OF A MACROMOLECULE
2
2 2 a hef
41~'
a
6ha
a
N
N~
l2
1 ah~r
2
3 aha
3aha
212
0
t~ 0
o
o
K,
K
2436:
M.L. POLVA~OV
the first place the large relative fluctuation should evidently be manifested in the kinetics of establishing phase equilibrium in the system. Galkin [9] shows that a similar situation arises in melting a polymer. In the case of weak sorption when ( V - Vc)/T<
lI
T R2
AF= - - ~ N ~ a ( V - Vc)+-~ ffff ,
(171
where R is the size of a coil, R 2 =Nora2~2, hof =aT/or ( V - V~) and li =~t2a/4. The first term in eqn. (17) describes the contact interaction between the monomeric linkages and the surface: Nldher is the number of units lying on the adsorbing surface, and (V-Vc) is the effective attraction experienced by an adsorbed unit (it depends on the competition between the attraction energy and entropy). Increase in stiffness results in a situation where art additional macroscopic parameter l~ appears, which describes the degree of cooperation in the bonding of the monomers to the surface in the coilintermediate state phase transition of the macromolecule. The second term in eqn. (17) describes the effective restriction to possible chain configurations, since the macromolecule is in an effective layer her. It can be obtained from a similar expression for an absolutely flexible polymer [2], if a transition is made to effective units of length lv. Moreover, the fact that increase in stiffness not ordy decreases the number of effective monomer units Nef=Na/l v, but also decreases her must be taken into account. The second re-normalization is associated with the fact that an effective unit has not two but (lv/a) contacts with the surface. This simple interpretation of the previously obtained equation for the free energy of a rigid chain is confirmed by accurate analysis of the lattice model, and is similar to that proposed by De Gennes [2] for flexible chains. Comparison of the equilibrium properties of flexible and rigid chains provides a means of revealing the effect of rigidity on the macroscopic characteristics of a macromolecule: the size of the layer in which .almost all the chain is found is decreased with increase in rigidity hef,~ -t, and the degree of sorption increases 0,-~a3. Equation (17) can be written in another form, i.e.
A~=
1 12~ (V--V,)2 N 12 ira T
(18)
Equation (18) is in agreement with a similar expression obtained by Birstein [6]. It is noted that eqns. (17) and (18) are comparable only whert the sorption term (V-Vc)/T<
Translatedby N. STANDEN
Mono- and bifunctional 2,4-dinitrophenyl derivatives of polyethylene oxide
2437
REFERENCES 1. R. J. RUBIN, J. Chem. Phys. 55: 4318, 1971 2. P. De GENNES, Idei skeilinga v fizike polimerov (The Skaling Concept in the Physics of Polymers). Moscow, 1982 3. T. M. BIRSTEIN, E. B. ZHYLINA and A. M. SKVORTSOV, Biopolymers 18: 1171, 1979 4. A. Yu. GROSBERG, Vysokomol. soyed. A24: 1194, 1982 (Translated in Polymer Sci. U.S.S.R, 24: 6, 1344, 1982) 5. A. M. KOSEVICH, A. S. KOVALEV and M. L. POLYAKOV, Svoistva zhestkikh makromolekul v adsorbiruyushchikh porakh (Properties of Rigid Macromolecules in Adsorbing Pores). Preprint number ITF-84-1P, Kiev, 1984 6. T. M. BIRSHTEIN, Vysokomol. soyed. A24: 1828, 1982 (Translated in Polymer Sci. U.S.S.R. 24: 9, 2088, 1982) 7. W. JACOBSON and W. W. STOCKMAYER, J. Chem. Phys. 18: 1600, 1950 8. L. D. LANDAY and Ye. M. LIFSHITS, Statisticheskaya fizika (Statistical Physics). Moscow, 1976 9. V. L. GALKIN, Studia Biophys. 87: 89, 1982
PolymerScienceU.S.S.R. Vot. 29, No. 10, pp. 2437-2444,1987 Printed in Poland
0032-3950/87$10.00+.00 © 1988 Pergamon Press pie
MONO- AND BIFUNCTIONAL 2,4-DINITROPHENYL DERIVATIVES OF POLYETHYLENE OXIDE* K. S. KAZANSKII, A. YA. KAMINSKII, N. V. PTITSYNA, V. S. ROMANOVA and I. N. TOPCHIEVA Institute of Chemical Physics, U.S.S.R. Academy of Sciences M. V. Lomonosov Moscow State University
(Received 14 May 1986) Mono- and bifunctional 2,4-dinitrophenyl derivatives of polyethylene oxide (PEO) are synthesized by two methods, i.e. by ionic polymerization of ethylene oxide and subsequent deactivation of the "living" polymers with appropriate reagents, and also by substitution of the OH groups of commercial polyethylene glycol (PEG). Polymers of M = 900-102,000 are identified by UV spectroscopy, chemical analysis methods, and from the molecular mass characteristics. Comparison of the two methods of synthesis shows that the first results in higher final 2,4-dinitrophenyl functionality. The synthesized polymers have immunogenic properties. FUNCTIONAL polymers based o n P E R with one or two e n d groups o f different character are available as preparative material a n d are receiving ever m o r e a t t e n t i o n . U p to the present a large n u m b e r o f p r o d u c t s of this type have beert synthesized [1-4], which c a n * Vysokomol. soyed. A29: No. 10, 2219-2225, 1987.
0032-3950/87 $10.00+.00 © 1988 Pergamon Press pie
THEORY OF THE ADSORPTION OF A MACROMOLECULE ON A PLANE SURFACE* M. L. POLYAKOV Physicotechnical I,-~stitute for Low Temperatures, Ukr.S.S.R. Academy of Sciences
(Received 12 May 1986) The adsorption of rigid macromolecules on a plain surface is analysed, and the rigidity of a macromolecule is described within the framework of a three-dimensional lattice model with a correlation belween ~he directions of adjacent monomers. The equilibrium characteristics of a macro:nolecule arc calculated, a~:d an ex!;ression for the free energy is derived. An increase in chain rigidity imparts a cooperative charader 1o the bonding of its monomers to the surface. Large fluctuations in the macroscopic values of the lengths of the adsorbed fragments and rings are a feature of the system. SIGNIFICANT success in studying the equilibrium properties o f absolutely flexible macromolecules interacting with a surface has been attained recently. A c c u r a t e results for a lattice model o f an ideal chain with an equilibrium distribution o f adjacent m o n o m e r s [1 ] have beert obtained. Qualitative interpretation o f the relation between the free energy o f a macromolecule and the adsorption energy [2] provides a means o f establishing the main laws, and also of allowing for the effect of th~ excluded vo!a~ne. A n o t h e r reasonable generalization o f the simplest model consists in allowing for the rigidity (this applies firstly to biomacromolecuies), i.e. the correlation between the directions o f the adjacent m o n o m e r s along the chain [3]. Nevertheless, a knowledge not only of the m e a n characteristics but also their fluctuations is necessary for a more complete understanding of the properties and special features of the system. In most works touching on the adsorption o f macromolecules fluctuations are not cortsidered. On the other hand, calculation o f the fluctuations of the m e a n values is o f interst for at least two reasons. Firstly, attraction of a macromolecule to a surface with a fairly high energy of interaction can result in the formation of a specific state. This state o f the macromolecule is, in a certain sense, intermediate between the state o f a coil and a globule, which differ in their fluctuation conditions [4]. Secondly, for c o m p a r i s o n o f the conclusions o f theory with the results of the corresponding experiments, for example by adsorptiort c h r o m a t o g r a p h y , only values with small fluctuations can be used. In this w o r k the special features of tb.e adsorption of a rigid m a c r o m o l e c u l e on a plane surface are studied, with. the object of analyzing the relation between the mean characteristics of the macromolecule a n d their fluctuations as a function o f the chain rigidity. The use of an accurately determined three-dimensional lattice molecule with. * Vysokomol. soyed. A29: No. 10, 2212-2218, 1987. 2430
Theory of the adsorption of macromolecule on a plane surface
2431
correlation between the directions of the adjacent monomers provides a means not only of calculating these values accurately, but also of expressing them in terms of a macroscopic parameter, i.e. the presistent chain length. Model. Statistical sum o f a macromolecule. It is suggested that all tire monomer units in the macromolecule have the same length a. Tire stiffness of the chain is manifested in the lact that configurations with different angles between adiacent monomers have different energies. The discussion will be restricted to the simplest case, when the configuratimml energy is equal to N-1
u= i=1
where tr~ is a unit vector directed along the i-th monomer, and [ is a parameter determiuing the excess chain stiffness. In a three-dimensional lattice model trz can have only six values:, i.e. ~rz--=+_n,., _+ny, ___n:, where n,, ny, and n: are the unit vectors of the coordinate axes. This model coincides with the model assumed by Birshtein cta/. I3]The quantity l determines the important macroscopic ch,.tracteri.~tic of the macromolecular coil, and the presistent length. The persistent length of the macromolccule 1~, is equal to the length along the chain for which a correlation betx~een the directions of the monomers disappears (is decreased by a factor of e). When I>'.T, i.e. for a rigid chain Ip=cca/4 , c~=exp(l/T) (2) The coordinate axes x, and y lying on the adsorbing surface will be selected. The interaction between the monomer units of the macromolecule with this surface can then be described by means of an external field V(z): I/(=)= =to when z < 0 (this condition corresponds to an absolutely impermeable face, since the proba~'lity of finding a unit at a point with a coordinate z is proportional to exp ( - V(z)/T)), K (z)= - l/ (V>0) at - = 0 , and V ( z ) = 0 at z > 0 , where the coordianates of the plane surface ==0, and k" is the sorption energy (an energetically more favourable unit is fouud on the adsorbing surface). Grosberg [4] used a shmlar external field for an:-t!ysis oi adsorption. Using the method proposed by Kosevich el al. [5] it is possible to find a clear expression tbr the statistical sum of the chain from N monomer tmlts with fixed z and z' coordinates and ~ and 6' directions of the initial a n d final units, ZN
,
~r)~j(z'-ag'n., -6')Bj,
=7N-t~e-ae°v-l)~j(z,
(3)
J where V= 1 + 4/'c(+ 1/e z,
E Eexp{v(a,,,~/r+ V(am-a~n.~/T} ~,~
B f ~=
m=O
a
(am, ~) ~ ( a m - a n : g , - g ) ; 2j and ~/~.(z, or) are the intrinsic values and corresponding
intrinsic functions of the equation q/j.(z, o') exp { - 2j + V ( z ) / T } = ~ a (6s) ~ j ( - - aon~, s) s
g (as) = ego e x p ( I 6 s / T ) ,
# o ~ = ez + 4~ + 1
(4)
M.L. POLYAKOV
2432
Accordingly, the problem ol determining the statisticalsum is reduced to the proplem of solving eqn. (4). ChemicaIpotential and size of effectivelayer. In order to analyse the equilibrium properties of a rigidmacromolccule it is necessary to solve the system of cquations (4) for the values ~ (z, ~), where ~ = &nx, +ny, ±nz. The minimum value of 2, which corresponds to the function ~o (z, ~), is of special interest,this decreasing exponentially on moving away from the adsorbing surface ~/o(z, o)- A (~) exp (- kz) at z I>a. The values of 2o and k can be found by analyzing the behaviour of ~o (z, a) at the boundary from eqn. (4) when z=O, a. Using ~o(z, ~) at z<0, it is not difficultto obtain such a system of equations I (.4-I) [.4-(~2- I)a0] [A-(~- I)29o]
2(c°shka-1)=Cdgo
A[A-(o~-l)(a+3)go]
Aexp(-ka)-(~2-1)go At At-(~+l)Zgo A - (a2 _ 1) g0 exp ( - ka)- 9o At + (3a + 1) ( ~ - 1) 0o
(5)
where A = exp (7- 2o), t = exp ( - V/T). It is not difficult to be convinced that the values of 20 and K determine the chemical potential of a monomer unit and the size of the effective layer respectively, in which almost the whole of the chain is concentrated. In fact, it follows from eqn. (3) that on approaching the dominating state of the chain [2] the free energy of the macromolecule, as calculated from the energy of the macromolecular coil F o = - N T I n 7, for N>>I is equal to
AF = F - F 0 = #N = T2o N ,
(6)
where/t is the chemical potential of the unit. The equilibrium density of the monomer units of the macromolecule irt this case is
p(z, ~)---NBotYo(z-aon,, -a)~o(Z,
•
(V(z)+V(z-aern,)]
~)exp t
~
j~
(7)
It follows from eqn. (7) that the contribution from the adsorbing surface is p (z, ~) ,-,exp ( - 2 k z ) . Since at a distance hcf h,f = (2k)- 1
(8)
the density of the units is decreased by a factor of e, it is natural to consider it equal to the size of the effective layer, in whichalmost the whole of the macromolecule is concentrated. A similar quantity is introduced by De Gennes [2] for an absolutely flexible chain. Distribution functions for an adsorbed section and a ring. When adsorbing surfaces are present the maeromoleeule consists of alternating adsorbed (contacting the surface) sections and rings joining adjacent adsorbed sections. The attraction of the monomer units to the surface results in a situation where at zero temperature the formation of an adsorbed monolayer is most favoured. On raising the temperature macromolecule sections are formed which do not contact (that is do not interact) with the surface, i.e.
Theory of the adsorption of' macromolecute on a plane surface
2433
the ring. This is because the entropy of the adsorbed section is less than that of a ring of the same length. To determine the equilibrium characteristics of the adsorbed section and ring it is convenient to use the method of a large canonical assembly [6]. The large statistical sum of the macromolecule can be represented in the form Z (/0 = {[ZI(/~)] -~ - Z"(/z)} - i,
,.
(9)
where Z l is the large statistical sum of the adsorbed section Z ' = go exp {(i, + V)/T} 1 + ( a - 1) (3a + 1) go exp {(#+V)/T_! 1 - ( a + 1)Zgo exp {(/2+ V)/T} '
(1 o)
and Z ~Jis the large statistical sum of the ring zU= ~ W(n)exp {fin/T},
(ll)
n= 1
where W (n) is the probability that the chain consists of n monomeric units, beginning in the plane z = a , and (n + 1)junctions make contact with the adsorbing surface only once. It follows from eqn. (11) that as a result of allowing for the formation of rings in the macromolecule long range entropy makes its appearance. Such long range action is not restricted to the ultimate n, and in the case of macromolecules of infinite length results in a phase transition [7]. The quantity W(n) determines the convergence of the series (11) and its derivatives with respect to /2. In its turn, the smoothness of the thermodynamic characteristics of the system, and consequently the existence and form of the phase transitions depend on the convergence of the given series. Unfortunately, direct calculation of the function W (n) involves formidable mathematical difficulties and has not been carried out even for a network model of an absolutely flexible polymer. A simpler method of studying the equilibrium characteristics of a ring consists in calculating the sum (11), i.e. the generating function for W(n), and in order to do this certain properties of the system under consideration are used. Firstly, the distribution function Z H is not clearly dependent on the sorption energy V. Secondly, the chemical potential of a unit of an endless chain N = oo can be found from the equation Z ~- Z u = 1 [6]. The given chemical potential will naturally be equal to the previously found chemical potential ,u, as given by eqn. (6). In order to determine Z t~, it is thus necessary to find, by means of eqn. (5), the values of V as a function of a and/2, and then to insert them in.{ZI} -1. After somewhat tedious but not complex transformations, the final result is ZH = exp ( - ka) - (a 2 - 1) go exp (p/T) (12) 1 - (a 2 -- 1) go exp ( - ka + p/T) where the value k is a function of/2 (system of eqn. (5)). Knowing the distribution function for the adsorbed section and the ring it is possible
2434
M.L. POLYAKOV
to find their mean length, and also the fluctuations of these values
li=ar-~lnZi(v), wi=ar
lnZ'(~)
I,,
(13)
where i= I, II for the adsorbed section and the ring respectively. In the remainder of this paper the discussion will be restricted to analysis and discussion of the following quantities as a function of the rigidity and sorption energy: the chemical potential p, the size of the effective layer h~f, the mean lengths of the adsorbed section Ii and the ring l~T,the relative fluctuations of these mean values w~ and ,wH, and also the number of boundaries between the adsorbed sections and. the rings
Nb = N { r ff---~[Z'(l~)+ Z"(p)]} - I
(14)
Analysis of results. For a thermodynamic description of the properties of the macromolecule it is necessary to solve the system of equations (5), i.e. to find the minimum inherent value of 20 and the parameter k, on which tire density of the monomers depends. An important feature of these equations will first be singled out, i.e. at the critical sorption energy (or critical temperature) Vc= - r In
cz2+ 2 e + 2 + ~ ( ~ 2 +4~+20) ~2 (c~/+4c~+ 1)
(15)
where 20 and k vanish. Moreover the energy of the macromolecule is equal to the energy of a macromolecular coil in free space. It is noted that under the given critical conditions of sorption the approximation of the doininating state of the chain is true only for N ~ o o . When V> Vc a state of the macromolecule in which the number of contacts with the surface N,a is proportional to N is more favourable, i.e. Nad=ON =
- N - dd# ~,
(I 6)
where 0 is the degree of sorption. When however V< V~ the chain is located mainly far away from the surface and (Nao/n)-~Owhen N ~ ~ . Because of the complexity of the system of eqn. (5) it is necessary to find a relation for the equilibrium characteristics of the molecule only in certain limiting cases. Table 1 gives a relation for these characteristics as a function of the rigidity and sorption energy when the chain rigidity is high, a>>l, and V~=2T/a. The values of the equilibrium characteristics of the macromolecule given in Table 1 bring out an interesting feature of the system under consideration. The fluctuations of certain macroscopic quantities, e.g. the number of adsorbed monomers Nad (Nao>>l), and the interface between an adsorbed section and a ring N~. (N~,>>I) are small and w,,~1/x/N, where N is equal to Nan or N~, respectively. Thus the behaviour of these quantities is the same as in most macroscopic systems [8]. On the other hand, although the adsorbed section and the ring are also macroscopic and l~>>a, /~>>a, their fluctuations are large even far away from the critical conditions of phase transition (15). In
<<
V-
1
~
3
V
a<
--<<--<< 1 T
1
a
lp<< ha~< - -
~<< ----<< ct Z
1
a
h~f~ f e
T
V - Vc
~t
1 --
Regions o f c h a n g e in a d s o r p t i o n energy
a2
i2h~f
alp
a
412
a
2ha
2
81~
2 2 a her
-- 12h2f "4 4lp her
alp
__
It T
I
'
~*
2 ( v - ~)
aT
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alp T
41p( V - V~)
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8
ah 2f
14 "P
a
tl
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4ha
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Ill
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R E S U L T S OF C A L C U L A T I N G THE EQUILIBRIUM CHARACTERISTICS OF A MACROMOLECULE
2
2 2 a hef
41~'
a
6ha
a
N
N~
l2
1 ah~r
2
3 aha
3aha
212
0
t~ 0
o
o
K,
K
2436:
M.L. POLVA~OV
the first place the large relative fluctuation should evidently be manifested in the kinetics of establishing phase equilibrium in the system. Galkin [9] shows that a similar situation arises in melting a polymer. In the case of weak sorption when ( V - Vc)/T<
lI
T R2
AF= - - ~ N ~ a ( V - Vc)+-~ ffff ,
(171
where R is the size of a coil, R 2 =Nora2~2, hof =aT/or ( V - V~) and li =~t2a/4. The first term in eqn. (17) describes the contact interaction between the monomeric linkages and the surface: Nldher is the number of units lying on the adsorbing surface, and (V-Vc) is the effective attraction experienced by an adsorbed unit (it depends on the competition between the attraction energy and entropy). Increase in stiffness results in a situation where art additional macroscopic parameter l~ appears, which describes the degree of cooperation in the bonding of the monomers to the surface in the coilintermediate state phase transition of the macromolecule. The second term in eqn. (17) describes the effective restriction to possible chain configurations, since the macromolecule is in an effective layer her. It can be obtained from a similar expression for an absolutely flexible polymer [2], if a transition is made to effective units of length lv. Moreover, the fact that increase in stiffness not ordy decreases the number of effective monomer units Nef=Na/l v, but also decreases her must be taken into account. The second re-normalization is associated with the fact that an effective unit has not two but (lv/a) contacts with the surface. This simple interpretation of the previously obtained equation for the free energy of a rigid chain is confirmed by accurate analysis of the lattice model, and is similar to that proposed by De Gennes [2] for flexible chains. Comparison of the equilibrium properties of flexible and rigid chains provides a means of revealing the effect of rigidity on the macroscopic characteristics of a macromolecule: the size of the layer in which .almost all the chain is found is decreased with increase in rigidity hef,~ -t, and the degree of sorption increases 0,-~a3. Equation (17) can be written in another form, i.e.
A~=
1 12~ (V--V,)2 N 12 ira T
(18)
Equation (18) is in agreement with a similar expression obtained by Birstein [6]. It is noted that eqns. (17) and (18) are comparable only whert the sorption term (V-Vc)/T<
Translatedby N. STANDEN
Mono- and bifunctional 2,4-dinitrophenyl derivatives of polyethylene oxide
2437
REFERENCES 1. R. J. RUBIN, J. Chem. Phys. 55: 4318, 1971 2. P. De GENNES, Idei skeilinga v fizike polimerov (The Skaling Concept in the Physics of Polymers). Moscow, 1982 3. T. M. BIRSTEIN, E. B. ZHYLINA and A. M. SKVORTSOV, Biopolymers 18: 1171, 1979 4. A. Yu. GROSBERG, Vysokomol. soyed. A24: 1194, 1982 (Translated in Polymer Sci. U.S.S.R, 24: 6, 1344, 1982) 5. A. M. KOSEVICH, A. S. KOVALEV and M. L. POLYAKOV, Svoistva zhestkikh makromolekul v adsorbiruyushchikh porakh (Properties of Rigid Macromolecules in Adsorbing Pores). Preprint number ITF-84-1P, Kiev, 1984 6. T. M. BIRSHTEIN, Vysokomol. soyed. A24: 1828, 1982 (Translated in Polymer Sci. U.S.S.R. 24: 9, 2088, 1982) 7. W. JACOBSON and W. W. STOCKMAYER, J. Chem. Phys. 18: 1600, 1950 8. L. D. LANDAY and Ye. M. LIFSHITS, Statisticheskaya fizika (Statistical Physics). Moscow, 1976 9. V. L. GALKIN, Studia Biophys. 87: 89, 1982
PolymerScienceU.S.S.R. Vot. 29, No. 10, pp. 2437-2444,1987 Printed in Poland
0032-3950/87$10.00+.00 © 1988 Pergamon Press pie
MONO- AND BIFUNCTIONAL 2,4-DINITROPHENYL DERIVATIVES OF POLYETHYLENE OXIDE* K. S. KAZANSKII, A. YA. KAMINSKII, N. V. PTITSYNA, V. S. ROMANOVA and I. N. TOPCHIEVA Institute of Chemical Physics, U.S.S.R. Academy of Sciences M. V. Lomonosov Moscow State University
(Received 14 May 1986) Mono- and bifunctional 2,4-dinitrophenyl derivatives of polyethylene oxide (PEO) are synthesized by two methods, i.e. by ionic polymerization of ethylene oxide and subsequent deactivation of the "living" polymers with appropriate reagents, and also by substitution of the OH groups of commercial polyethylene glycol (PEG). Polymers of M = 900-102,000 are identified by UV spectroscopy, chemical analysis methods, and from the molecular mass characteristics. Comparison of the two methods of synthesis shows that the first results in higher final 2,4-dinitrophenyl functionality. The synthesized polymers have immunogenic properties. FUNCTIONAL polymers based o n P E R with one or two e n d groups o f different character are available as preparative material a n d are receiving ever m o r e a t t e n t i o n . U p to the present a large n u m b e r o f p r o d u c t s of this type have beert synthesized [1-4], which c a n * Vysokomol. soyed. A29: No. 10, 2219-2225, 1987.