J. CHEM. SOC. FARADAY TRANS., 1994, 90(6), 889-893 Theory of Monolayers of Non-Gaussian Polymer Chains grafted onto a Surface Part 1.-General Theory Victor M. Amoskov and Victor A. Pryamitsyn Institute of the Problems of the Mechanical Engineering, 61 Bolshoy pr., V.O., 199178 St.-Petersburg, Russia A theory describing layers of polymer chains grafted to a flat surface (polymer 'brush') is developed. We consider a brush of chains with arbitrary extensibility and, within the framework of molecular field theory, we suggest a general scheme for calculating the selfconsistent pseudo-potential, which determines the structure of the brush. This potential appears to be defined only by the mechanism of extensibility and is independent of the interactions between chains in the brush.A model potential for freely jointed chains (FJC) was calculated and used for describing an FJC polymer brush in good solvent conditions. We found good agreement between our results and the numerical calculations of Skvortsov eta/. The properties of polymers adsorbed at solid/liquid or liquid/ liquid interfaces are important in areas such as tribology and biophysics. One type of adsorption is 'grafting' or end adsorption, in which polymer chains are attached by one end to a surface. This may be achieved in several ways, for example, by constructing a block-copolymer in which one block is strongly adsorbed onto the surface. Similar struc- tures appear in the superstructures of block-copolymers. The resulting structure is usually called a 'brush'.The theory of brushes has been developed by many authors. I-' Initially, polymer brushes were analysed using the scaling method by Alexander,' De Gennes2 and Birshtein and Zh~lina.~ The general result of these theories was that the polymer chains in the brush are strongly stretched. To explain this result, let us consider a system consisting of long polymer chains grafted, by one end, to a flat surface. The number of segments in the chain, N, is one of the major parameters in the theory: N + 1. It is known that under the conditions of negligible inter- action between segments, the distance between the free ends of the chain is (hi) = Nu2, where u is the length of the Khun segment. In the presence of repulsion, with a finite radius of interaction between the segments (good solvent conditions), (h,) K N', where v z0.6." Let us estimate the size of the chains in the 'brush' in the presence of repulsion.Since the density of segments in a brush of long chains has to be finite, the height of the brush, Hb, has to be proportional to N if the grafting density per unit area, 0, is constant. This dependence for the planar 'brush' is universal in any solvent or in a melt. Hence the chains in a planar brush are strongly stretched with respect to their ideal size, Hdh, a No.', and their size in a good solvent Hdh, a This stretching of chains in the brush is caused by the repulsion between chain segments. Scheutjens and Fleer4 employed self-consistent field (SCF) theory for the lattice model of the polymer brush.They calcu- lated numerically the structured details of brushes of finite N. Later Skvortsov et ul.' used the same model and combined it with the original procedure for extrapolation of the results of numerical calculations for finite N to N + a. The analytical SCF theory of polymer brushes was elabo- rated by Semenov6 for the case of brushes without solvent; it was developed for brushes with solvent by Pryamit~yn,~ and Zhulina et uL8 and independently by Milner et u1.9*10This theory uses the Gaussian model of polymer chains," and, as is shown in ref. 5, it appears to be an asymptote of the Scheutjens-Fleer model for the case where N -+ 00 and Hb 4 Nu. (contour length of polymer chains).Unfortunately, the Gaussian model of polymer chains is inadequate for strongly stretched chains, when Hb < Nu. Such strong stretching of polymer chains can appear in the charged brushes of polyelectrolyte chains, or in very dense grafted brushes, for example, in block-copolymer super-structures, when one of the blocks is a flexible polymer and the other is a rod-like mesogenic polymer. Shim and Cates constructed a very elegant approach for polymer-chain brushes with the ad hoc approximation of finite extensibility of the polymer chains.12 They found qual- itative agreement with Scheutjens and Fleer's numerical cal- culations, but no quantitative agreement. In our opinion, this is due to the fact that Shim and Cates' (SC)ad hoc approx-imation of finite extensibility cannot be realized in a physical model of the polymer chain (see below).In this paper, we discuss general SCF theory applied to brushes with an arbitrary mechanism of chain extensibility and apply this theory to the model of freely jointed polymer chains, which seemed to be adequate for polymer chains with finite extensibility. To compare our theory with the results of numerical calculations we examine dense grafted polymer brushes in good solvent conditions. Brushes of polyelectrolyte polymers and mesogenic brushes will be discussed in the future papers. Model In the SCF (or molecular field) model an assembly of inter- acting chains is described as a group of non-interacting chains in the self-consistent pseudo-potential, pLcp(r)],where p(r) is the local density of polymer-chain segments and p@) is their chemical potential.In thermodynamically stable systems, p(p) is an increasing function of p. If the grafting density is relatively high, NuZ% u-l, one can assume that the brush is homogeneous along the grafting surface and p(r) = p(x), where x is the distance from the graft- ing surface. Thus the problem of describing an assembly of chains in the brush is reduced to that of one chain in a one-dimensional, self-consistent stretching field. A long polymer chain in a non-uniform stretching field can be described as a spring in the stretching field." The polymer-chain conformations with maximum statistical weight correspond to the state of static equilibrium of the spring in the stretching field, whereas the extensibility of the spring corresponds to the local extensibility of the chain.It is significant that to produce chain stretching, p(x) has to be a decreasing function, consequently p(x) also has to be a decreasing function. It is reasonable to use the average elon- gation (6x(f)) of a chain of N segments under a stretching force,f, applied to the ends of the chain for the definition of the extensibility of a polymer chain, e(f): where F(f) is the Gibbs energy of a chain of N segments under a stretching force,f, applied to the ends of the chain. It can be proved that for any physical model of polymer chains, F(f)has to be an analytical even function off: Note that the SC model does not meet this condition.The system of equations describing the static equilibrium of the nth segment of a polymer chain of N segments in a field of stretching force,f(x), via the approach of a non-linear spring in a one-dimensional stretching field with potential p(x),in dimensionless form is: where 1 = n/N, z = x/Na, p = uf/kT (dimensionless stretching force),f(z) = aV(z)/az, V(z) = -p/kT (dimensionless increas- ing function of z), e@) is the non-linear extensibility of the chain, k is the Bolzmann constant and Tis the temperature. The boundary conditions are: Z(~)I~=~ = 0; p(I)(l=l= 0. The first integral of eqn. (2) is E(p) = V(ZJ -V(Z) (3) where E(p) = SpO e(p’)dp’ and z, is the height of the free end of the spring.The value of z, is determined by the equation: dzltA[V(z,) -V(z)] = (4) where A(V) = e[@(V)], E[qV)] = V. Eqn. (3) and (4) give solutions to eqn. (2), if the potential, V(z), and extensibility e(p), are known. For models of freely jointed polymer chains in a three- dimensional continuum and on a simple cubic lattice, the non-linear extensibilities, Ej(p),ej(p) and Ejl(p),ej,(p), are : sinh pEj@) = log( F), ej(p) = coth p -p-1 Selfconsistency Equations for the Polymer Brush To calculate the self-consistent potential we have to calculate the local density of polymer segments in the brush. From eqn. (2) we can define the local density of segments for a spring in stretching field, V(z): Qualitative analysis shows that p(z, 2,) is a monotonic increasing function of z for any stretching field, V(z),and has a singularity at the point z = z,. As mentioned above, to produce stretching of chains in the brush the local density of J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 chain segments in the brush has to be a decreasing function of z. How can we increase p(z, z,) for each chain in the brush and maintain the necessity of decreasing the brush density p(z)? Let us consider a polydisperse brush of chains with all possible lengths 0 < L < Lo. Let w(9)(9= L/Lo) be the distribution function of the length of the chains and g[z,(9)] the distribution function of the height of the chain ends, where ze(9)= x,/Lo is the reduced height of the end of the chain with length 3,the function Y(z,) being the length of the chain with end height z,.They are connected by the equation : (7) From eqn. (4) we have : If p(p), w(9) and cr are known, eqn. (7)-(9) completely determine the brush parameters. The selection of the distribu- tion z,(9) provides self-consistency for the potential V(z). It is evident that z,(9) has to be an increasing function of 9. To produce decreasing p(z) the free ends of the polydisperse chains have to be distributed from z = 0 to z = z,(I). It is easy to prove from eqn. (8) that if an excluded zone, where the free ends of the chains are absent, exists between z = 0 and z,(l), the function p(z) would be increasing in this zone. Evidently we have a similar situation for monodisperse brushes.The only possibility for providing a continuous dis- tribution of the free ends of the monodisperse chains in the brush is to use the potential in which the spring is in a state of indifferent equilibrium. The equation for this potential is : If the dependence of the chemical potential, p(p), is known we can calculate the distribution of the density of polymer segments in the brush: -p[p(z)] = kTV(z) + constant } (11)p(z) dz = O/Ulmax where z,, is the maximum possible height of the chain seg- ments. For the free brush with local interactions, without external restrictions, the value of zmax is determined by the =requirement ap/ap Iz=zrmx0 and by the normalization requirement for the density, eqn.(1 1). Eqn. (8) is transformed into the equation for self-consistency : p[ V(z)] = c Izmaxp(z,z’)g(zf) dz’ g(z’) dz’ where g(z) is the distribution function of monodisperse free chain ends, which is independent of the description of mono- J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 891 disperse brushes, and p(V) is determined from the equation for the chemical potential, V = -p(p)/kT. If V(z) is the solution of eqn. (lo), we can invert eqn. (12) (see Appendix): g(2) = --b{V[~(Z)] -V(Z'))dz' (13) where V[Z(z)] = V(z,,.J -V(z), $(v) = (dp(v)/du). For small z,,, where [V(z,) -V(z)] is also small, the Gaussian approach is correct, i.e. &I) zp/3, A[V(z,) -V(z)] z J[V(z,) -V(z)] and eqn. (10) and (12) are trans- formed to: This is the well known system of equations for a brush of Gaussian The pseudo-potential for indifferent equilibrium of a Gaussian chain is V,(z)= 3n2/8 z2 = 3/2(nx/ ~QN)~.Self-consistent Potential for the Model of Freely Jointed Chains We expand y(z) and Vjl(z) in a power series of z: q(z) = IF=Vf)z2& and vjl(z) = c;= V~')Z~~,where Vy)= (3/8)z2, Vt)= (9/320)n4, Vg)= (153/56000)~~, Vt)= (3321/12544000)~~; Vt)= (1535643/60368000000)n'O, vp = (849O22533/3453O496O)n12, and so on; V?') = (3/8)n2, V(Zjl)= 0, VC,jl)= (9/2560)n6, Vv" = ( -81/114688)n8, V:') = (5769/22937600)n'O and so on (see Appendix); the first 30 values of Vf) are presented in Table 1. It is significant that Vg')= 0; this is because Ejl(p)= p2/ 6 + O(p6),i.e. the model of freely jointed polymer chains on a cubic lattice at light extension is closely allied to the Gauss- ian model.In general, except for the simple cubic lattice, E(p) = p2/6 + O(p4)and V(z) = (3n2/8)z2+ O(z4). Table 1 First 30 values of Vf)and B, 1 3.701 101 650 3.701 101 650 2 2.739 630 685 -0.961 470 965 3 2.626 652 618 -0.112 978 067 4 2.51 2 070 432 -0.114582 186 5 2.382 221 849 -0.129 848 584 6 2.272 557457 -0.109 664 392 7 2.200499 532 -0.072 057 925 8 2.162403637 -0.038 095 895 9 2.144 171 977 -0.018231 660 10 2.131 963 619 -0.012208358 11 2.1 18 021 735 -0.013941 884 12 2.101 168 777 -0.016 852957 13 2.084 057 663 -0.017111 114 14 2.069 843 538 -0.014214 125 15 2.060 092 646 -0.009750 892 16 2.054 408 293 -0.005 684 353 17 2.05 1 234 597 -0.003 173 696 18 2.048 948 605 -0.002 285 992 19 2.046 572 565 -0.002 376 040 20 2.043 897 907 -0.002 674 658 21 2.041 191 961 -0.002 705 947 22 2.038 801 424 -0.002 390 5 37 23 2.036 894 352 -0.001907 072 24 2.035 415 417 -0.001 478 935 25 2.034 189 192 -0.001 226 224 26 2.033 054 638 -0.001 134555 27 2.03 1 943 401 -0.001 11 1237 28 2.030 878 752 -0.001 064 649 29 2.029 924 035 -0.000954717 30 2.029 126 557 -0.000 797 477 1 I 20.00 15.00 ! 5.00--$ ,,,*' ,,,' Z Fig.1 Self-consistent pseudo-potential, q(z),for a polymer brush of FJCs in a good solvent (-); self-consistent pseudo-potential, VJz), for a polymer brush of FJCs on cubic lattice (---); parabolic self- consistent pseudo-potential, V,(z), for the Gaussian brush (-.-); pseudo-potential for Shim and Cates' model' (-- - - -) We analyse the continuum model of FJCs in more detail because it seems to be more realistic for the applications of this theory. c(z) is singular when z + 1. An analysis of this expansion shows that the main part of this singularity is: vj(z)Oc 2(1 -Z2)+ + 0[(1 -z2)-'] This allows us to estimate V,(z): 30 2z2(1 -Zz)-l + c A&z2& 30 < y(z) < (1 -Z2)-l 1Bkz2& &= 1 &= 1 (14) where A, = V, -2 and B, = V, -V,-'. The coefficients V, and B, are presented in Table 1. The expansion of eqn. (14) converges quite well.The accuracy of this estimation is <0.001% on z = 0.9 and 0.4% on z = 0.99. We use this expansion for the calculation of the V,(z) dependence, which is shown in Fig. 1. Dense Grafted Brush in a Good Solvent The usual model for the description of polymers in solutions is the Flory-Huggins lattice model.' ' For an athermal good solvent the chemical potential, pFH, for this model is : The density of polymer segments in the brush for this model is: The value of zmax is determined from the normalization con- dition : Zmax -lmXexp[q(z)-Y(z,,,)] dz = ou2 (1 7) The dependences of p(z) for different au2 and z,,,(m2) are presented in Fig. 2 and 3. One can see that if ou' < 0.1, the Gaussian approach is quite adequate, the density profile is parabolic; if m2> 0.5, the density profile is flattened; and if au2 > 0.8, the density profile is step-like and z,,, x au2.Eqn. (13) allows us to cal- 1 .oo 0.80 0.60 b.. h h(v P 0.40 0.20 Z Fig. 2 Density profiles for the polymer brush of FJCs in a good solvent (lattice model) for ou2 =0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9 culate the distribution of ends, g(z): 1 d5g(z) = --exp[vj(z) -vj(zmBz) exp[vj(z')] dz' (18)au2 dz where V(2) = V(z,) -V(z); the function, g(z), for different au2 is presented in Fig. 4. One can see that in the region au2 x 0.5 the behaviour of g(z) is changed. If auZ< 0.5, g(z) is a convex function and the maximum value of g(z) decreases with increasing aa2; if nuz >0.5, the maximum value of g(z) increases with increasing au2.Comparison with Other Models Skvortsov et a1.' used the SCF theory of Scheutjens and Fleer for a brush of FJCs on a three-dimensional cubic lattice. They calculated numerically the structural details of brushes of finite N and applied a special procedure for the extrapolation of these results to N +co.The data of Skvort- sov et a1.' for the dependence z,,(aa2) and our calculations 1.o 0.8 0.6 0.4 0.2 I0.0 I"~'"'~~I""""'I"""'~'l""''~''I''~''"''I 0.0 0.2 0.4 0.6 0.8 1 .o oa Fig. 3 Maximum height, z, vs. surface fraction of polymer ou': (-) polymer brush of FJCs in a good solvent; (..) polymer brush of FJCs on a cubic lattice; (--) Gaussian brush in a good solvent, (-.-) asymptotic z,, = [40u2/7r2-j1'3; (---) Shim and Cates model;12(A) data of Skvortsov et uf.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 30 -h-3 20-b: 10i, z Fig. 4 Distribution function of free ends: (-) g(z) for a polymer brush of FJCs in a good solvent (lattice model) for oa2 = 0.1,0.2, 0.3, 0.4,0.5,0.6,0.7,0.8,0.9; (---) g(z) for a Gaussian brush, au2 = 0.77 for the cubic lattice model are shown in Fig. 3. One can see a good agreement between these results and our theory. Note that if au2 c 0.5, the results for this model and the Gaussian model are closely allied. To compare our results with these of Shim and Cates we reduced their self-consistent potential' to the form : z(V) = 1 -exp -erfc371 Fig. 1 shows this potential.One can see that this potential is not an even function of z: 3n2 3n3VAZ) = -z2 i--z3 4-* * 8 16 The dependence of zm,(au2) for the SC model is presented in Fig. 3. The presence of the unphysical term z3 in the potential causes a strong divergence of the SC results from the results of other models. This compelled Shim and Cates12 to use dif- ferent phenomenological parameters in order to compare their results for brushes of chains with finite extensibility with the results of the more detailed self-consistent Scheutjens- Fleer model4 for different grafting densities. We thank July Lyatskaya, Ekaterina Zhulina, Sergey Frid- rikh and Alexander Skvortsov for very helpful discussions. This work was supported, in part, by a Soros Humanitarian Foundation Award by the American Physical Society.Appendix To solve eqn. (10) we introduce new variables: u = V(z'), u = V(z), U = V(zmax), dz =(dz/du) do. We have: To solve eqn. (Al) we can use the Laplace transformation: P(o)= rexp( -oz)F(z) dz After applying the Laplace transformation to eqn. (Al) and returning to the independent variable z in the first integral we J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 have : exp[ --uV(z)]dz J exp[ -oE(p)] dp = (A2) 0 0 As with the usual models of polymer chains with finite extensibility, E(p) = tion of z, V(z) = Lrn=Ekp2‘, then V(z)is an even func- V, z”. To calculate V, we expand both the integral in eqn. (A2) by the pass method in the power series on U-’/~-”,n 2 0 and multiply both series.After comparing the corresponding coef- ficients we can calculate V,. Since polymer chains must exhibit Gaussian behaviour under very small extensions, E, = 6, V, = in2.We have written a computer program for calculating these coefficients. To solve eqn. (12) we introduce new variables, u = V(z’), u = V(z), U = V(z,.,-), u = V(z’), dz = (dz/du) du. We have: q(u) dup(u) = IU-A(u-U) where 4(u)= g[x(u)] (dz/du). To solve eqn. (A3) we introduce w = (U-u), R(w) = p(U -u) and Q(u) = q(U -u): After applying the Laplace transformation to eqn. (A4) and using eqn. (A3) we have: dz d(w)[exp( -mu) -du du = Q(o) 045) After inverting the Laplace transformation, eqn. (A6) follows: dR(w -U) dz -duQ(w) = du du and by returning to initial variables we obtain eqn.(13). References 1 S. Alexander, J. Phys. (Paris), 1977,38,977. 2 P-G. DeGennes, Macromolecules, 1980,13, 1069. 3 T. M. Birshtein and E. B. Zhulina, Vysokomol. Soedin., Ser A, 1983,25,1862. 4 J. M. Scheutjens and G. J. Fleer, J. Phys. Chern., 1979,83,1619. 5 A. M. Skvortsov, A. A. Gorbunov, I. V. Pavlushkov, E. B. Zhulina, 0.V. Borisov and V. A. Pryamitsyn, Polym. Sci. USSR, 1988,30,1706. 6 A. N. Semenov, Sou. Phys. JETP, 1985,61,733. 7 V. A. Pryamitsyn, The Orientational Ordering of Polymer Systems, Candidate Thesis, Institute of Macromolecular Com- pounds, Leningrad, 1987. 8 E. B. Zhulina, V. A. Pryamitsyn and 0.V. Borisov, Polym. Sci. USSR, 1989,30,205. 9 S. T. Milner, T. A. Witten and M. E. Cates, Macromolecules, 1988,21,2610. 10 S. T. Milner, Science, 1991, 251,905, and references therein. 11 P-G. DeGennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1979. 12 D. F. K. Shim and M. E. Cates, J.Phys. (Paris), 1989,50,3535. 13 T. M. Birshtein and 0. V. Ptitsyn, Conformations of Macro-molecules, Wiley-Interscience, New York, 1966. Paper 3/04708C; Received 4th August, 1993