Langmuir 1994,10, 2281-2288
2281
Adsorption of Homopolymers on a Solid Surface: A Comparison between Monte Carlo Simulation and the Scheutjens-Fleer Mean-Field Lattice Theory Yongmei Wang and Wayne L. Mattice* Institute of Polymer Science, The University of Akron, Akron, Ohio 44325-3909 Received September 20, 1993. In Final Form: April 28, 1994@ A comparison of a multichain Monte Carlo simulationof homopolymer adsorptionon a solid surface with the mean-field lattice theory of Scheutjens-Fleer is presented. The comparison reveals certain systematic deviations between the theory and the simulation, which are reflected in the bound fraction of the chain, surface coverage, and the adsorbed amount. Such deviations can be attributed to two approximations adopted in the theory. One approximation is the allowance of direct back-fold of the chain, and the other approximation is the random-mixingwithin each layer. The allowance of direct back-fold of the chain is a result of treating the chain as a Markovian chain. It gives rise to a difference in the number of allowed conformations compared to the Monte Carlo simulation. However such differences do not affect the distribution of chain segments in homogeneous solution. It would only cause a difference when the chain is in an inhomogeneous solution or when it encounters an impenetrable solid surface. The study reveals that the deviation in bound fraction introduced due to the allowance of direct chain back-fold persists throughout the whole range of concentration. It is more pronounced at weak adsorption. On the other hand, the random mixing approximation works better in moderate concentrationunder weak adsorption since the adsorbed chains can more easily penetrate each other. In the strong adsorptionlimit, the adsorbed chains are confined to two dimensions and they resist interpenetration. Thus the deviation in surface coverage and adsorbed amount caused by the random mixing between the theory and simulation is more pronounced under strong adsorption.
Introduction The adsorption of homopolymers on a surface has been under extensive study since early 1930 due to its importance in numerous applications. The earlier theoretical work was mainly concerned with a single long flexible chain adsorbed on the s u r f a ~ e . ~The - ~ segment-excluded volume effect is either ignored or treated approximately. Rigorous treatment of oligomers of up to four segments has been presented,1° but it has not been applied to long polymers due to the tremendous computational difficulties involved. A real advance in the theory of polymer adsorption on a surface was marked by the advent of meanfield lattice models of polymers at an interface, represented mainly by the work of Scheutjens and Fleer.11-13 In the theory of Scheutjens and Fleer, the assumption that segments located in any part of the chain contribute equally to the segment density at any distance from the surface is avoided. Instead it is determined selfconsistently. Thus they have accurately predicted the distribution of tails, trains, and loops for the adsorbed chain on the surface. The theory has also been extended to the adsorption of block and random copolymer^.'^-'^ Abstract published in Advance ACS Abstracts, June 1, 1994. (1) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: London, 1983. (2) Simha, R.; Frisch,H. L.; Eirich, F. R. J.Phys. Chem. 1953,57, 584. (3)Frisch, H. L. J.Phys. Chem. 1966,59, 633. (4) Silberberg, A. J. Phys. Chem. 1962,66, 1872, 1884. (5) DiMarzio, E. A.; McCrackin, F. L. J.Chem. Phys. 1966,43,539. ( 6 ) Rubin, R. J. J. Chem. Phys. 1965,43,2392. (7) Hoeve, C. A. J. J. Chem. Phys. 1968,44, 1505. (8) Silberberg, A. J. Chem. Phys. 1967,46, 1105. (9) DiMarzio, E. A.; Rubin, R. J. J. Chem. Phys. 1971, 55, 4318. (10) Ash, S. G.; Everett, D. H.; Findenegg, G. H. Trans.Faraday Soc. 1970, 66, 708. (11)Scheutjens, J. M. H. M.;.Fleer, G. J. J.Phys. Chem. 1979,83, 1619. (12) Scheutjens, J. M. H. M.; Fleer, G. J. J.Phys. Chem. 19SO,84, 178. (13) Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1985,18, 1882. @
Along with the theoretical developments, Monte Carlo s i m u l a t i ~ n l ~has - ~ ~also played an important role in understanding the adsorption of polymers. Most of the earlier simulations deal with a single chain case, in which the effect of segment-excluded volume interactions has been studied. More recently there have been some multichain Monte Carlo simulationsz3 of adsorption on the solid surface. However due to limitations inherent in the model, a rigorous comparison of Monte Carlo simulation results with the theory of Scheutjens and Fleer has not been made. Such a Comparison is nevertheless of importance, because the theory has adopted a few approximations which are used in many theoretical studies, and the validity of such approximations has not been carefully examined. We present a Monte Carlo simulation study of homopolymer adsorption on a solid surface. Our interest is focused on the comparison between the simulation results and the theoretical calculations. The comparison reveals some interesting aspects which are caused by the approximations introduced in the treatment by Scheutjens and Fleer. Recently Smith et a1.%performed a Monte Carlo simulation study of a bulk polymer melt between two plates, and the results were compared with the same theoretical calculation as extended to the bulk polymer (14) Evers, 0.A.; Scheutjens, J. M. H. M.; Fleer, G. J.Macromolecules 1990,23,5221. (15) Evers, 0.A.; Scheutjens, J. M. H. M.; Fleer, G. J. J.Chem. SOC., Faraday Trans. 1990,86, 1333. (16) Evers, 0.A.; Scheutjens, J. M. H. M.; Fleer, G. J.Macromolecules 1991,24, 5558. (17) McCrackin, F. L. J. Chem. Phys. 1967,47, 1980. (18) Lax,M. J. Chem. Phys. 1974, 60, 2245. (19)Lax, M. J. Chem. Phys. 1974, 61, 4133. (20) Mark, P.; Windwer, S. Macromolecules 1974, 7, 690. (21) Clayfield, E. J.; Lumb, E. C. J.Colloid Interface Sci. 1974,47, 6. (22) Clayfield, E. J.; Lumb, E. C. J.Colloid Interface Sci. 1974,47, 1s.
(23) Balazs, A. C.; Huang, K.; McElwain, P.; Brady, J. E. Mucromolecules 1991,24, 714. (24) Smith, G. D.; Yo0n, D. Y.;Jaffe, R. L. Macromolecules 1992,25, 7011.
0743-746319412410-2281$04.50/00 1994 American Chemical Society
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2282 Langmuir, Vol. 10, No. 7, 1994 interface problem. Our study is complementary to their study, with emphasis on the adsorption from a dilute to semidilute solution. In the following section, we will first give a brief description of the theory of Scheutjens and Fleer, specifically indicating the approximations that have been adopted in their theory. More detailed information can be found in the original references.11J2 The Monte Carlo simulation model will be described in the same section. The results and comparison between the Monte Carlo simulation and the mean-field lattice theory will be presented in the following section, along with some discussion.
Models Pertinent Aspects of the Scheutjens-Fleer Mean-Field Theory: The theory of Scheutjens and Fleer is based on a multilayer lattice model with an impenetrable adsorbing surface at the zeroth layer. Perpendicular to the adsorbing surface layer, the segmental density distribution is heterogeneous. Parallel to the layer, the segments are assumed to be random mixed. The nearest neighbor sites of a position on the lattice are distributed such that a fraction of A1 lies in each adjoining layer (for simple cubic lattice, A1 = l/,J and a fraction oflo on the same layer. One of the central quantities of the theory is the probability a free segment will find itself in layer i, with reference to a bulk polymer solution of concentration 4..
Here @O(i)is the solvent volume fraction in layer i, and
4: is the solvent volume fraction in the bulk. The factor $O(i)/& arises from the segmental excluded volume, and it follows directly from the random mixing assumption. The probability of a segment entering into layer i is proportional to the number of vacant sites in that layer. When layer i is completely filled, r$O(i) is zero, and pi is therefore zero. The two exponential parts are due to the difference in enthalpic energy between layer i and the bulk. The &,i is one fori = 1and zero for all other values. xsis the reduced energy when a segment is in direct contact with the surface, andx is the reduced solvent and segment interaction energy. ($(i)) is equal to
Mi))= &[4(i - 1)+ 4(i + 111 + Ao$(i)
(2)
Thus the first exponential factor is also a result of the random mixing assumption. A segment in layer i would have 111 fraction interactions with the solvent in layer i 1and i - 1, and ;lo fraction interactions with the solvent in layer i. pi defines the distribution of free segments in successive layers. The polymer chain is constructed byjoining r segments indexed by s = 1, 2, ...r. The probability that an end segment resides in layer i is a sum of the probabilities of a previous segment in either the same layer, or the two adjacent layers, weighted bypi. Lettingp(i,r) denote the probability of the end segment of an r-mer in layer i, then
+
+
p(i,r) =p,[Ag(i - 1, r - 1) A,p(i
+ 1, r - 1)+ Ag(i,r - 111 (3)
Here lies another implicit approximation. The chain is modeled as a Markovian chain. The probability distribution of the end segment is only a function of the position of the previous segment. The direct back-fold of the chain is not prohibited. Acorrection to this approximation would
require the knowledge of the distribution of three successive segments. The above idea can be further applied to successive segments and simply expressed using the matrix formulation:
P(r)= WP(r - 1)= W' - ~ ( 1 )
(4)
P(r), P(r - 1)and P(1)are vectors of M dimension (M is the total number of layers) denoting the end segmental probability in all the layers. W is a M x M matrix, with elements equal to
w 1J, =;I. . J-Pi
(5)
The probability of the sth segment of an r-mer in layer i is then considered as two random walks with steps of s - 1and r - s, respectively, and both end segments of such walks end at layer i. The corresponding end segment probabilities are p(i,s) and p(i,r - s 1). Thus
+ p(i,s,r) = p(i,s)p(i,r - s + l)/pi
(6)
The segmental density profile @(i) is just the summation of the probability of all segments in layer i times the reference concentration.
(7) These equations then determine the distribution of the homopolymer on the surface while in equilibrium with a bulk phase of concentration $*. The properties, such as the amount of adsorbed homopolymer, r, and surface coverage, 8, can be uniquely determined since the probabilities of the segments in each layer are known. In summary, there are three approximationsintroduced in the theory, namely random mixing within each layer in accounting for the prohibition of double occupation, random mixing in each layer in accounting for the enthalpic interaction, and the allowance of direct backfold ofthe chain. In the present study, the segment solvent interaction is zero. Thus only the effect of the first and third approximations will be explored. Monte Carlo Simulation: The Monte Carlo simulation is carried out with a simple cubic lattice of size L, x Ly x L,,where L,=Ly = 25-50 and L,= 40-80 with periodic boundary conditions applied in the xy plane only. The first layer z = 1is taken as the adsorbing layer. The last layer z = L,is a noninteracting solid wall. The choice of L, is based on the chain length. L,must be large enough (at least 10 times the radius gyration of the chains) so that it will not affect the equilibration between chains adsorbed on the surface and those in the bulk phase. The reduced interaction energy of a segment with the surface, xs = uJkT, is assigned whenever a segment is in direct contact with the adsorbing layer. This is the same one as adopted in the theory of Scheutjens and Fleer. The polymer chain is represented by connected beads on the lattice, and no site can be occupied by more than one bead. The vacancies represent the solvent molecules. For the simulation reported here, the segment has no interaction with the solvent molecules. The solvent is an athermal solvent. The simulation starts with a fixed number of chains in the box. The motions of the chain that are used to equilibrate the system are reptation and the extended Verdier-Stockmayer relaxation. The chains are first randomized with all the energies equal to zero. Then the desired xs is applied, and the system is equilibrated using
Adsorption of Homopolymers on a Solid Surface
Langmuir, Vol. 10, No. 7, 1994 2283
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Figure 1. The volume fraction of segments in each layer after equilibrium. The system has 100chains ofN= 10 andk = 1.0. The lattice has L, = Ly= 25, L, = 40. the Metropolis rule. After equilibration, if a polymer chain has any nearest neighbor in contact with the adsorbing wall, this chain is considered as being adsorbed. Otherwise the chains are considered as in the bulk. The adsorbed amount, l-' is then equal to
r = (n,)NILJy
(8)
where (n,) is the average number of chains that are adsorbed on the wall and N is the number of beads in the chain. This definition of r is also adopted in the theory of Scheutjens and Fleer. There are two more quantities that have been monitored in the simulation. One quantity is the average bound fraction P of the polymer chains adsorbed on the surface.
P = (n,)/N
(9)
where (ni)is the average number of segments per adsorbed chain that are in direct contact with the wall. Another quantity is the surface coverage, 8, which is the fraction of the wall that is directly covered with a segment. 8, P, and r satisfy the relationship
e=py
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z
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Since the theory of Scheutjens and Fleer is also based on a multilayer lattice model, all these quantities have a one-to-one correspondence between our Monte Carlo simulation and the mean-field lattice theory.
Results and Discussion In order to compare the properties that can be monitored in the simulation with the calculation according to the Scheutjens and Fleer's theory, we need to specify the bulk concentration (denote as hereafter) after the system reaches equilibrium. This specification is done by calculating the average segmental density profile along the z direction aRer the system reaches equilibrium, with a typical result depicted in Figure 1. The horizontal axis is the z coordinate, and the vertical axis is the average volume fraction of segments in each layer. At z = 40there is another solid wall which has no interaction with the segments. As long as that solid wall is far enough from the adsorbing layer, it should not affect the adsorption of the polymer chains on the adsorbing surface at z = 1.The C#J is determined by averaging the density profile over the range of z where there is no tendency for an increase or
Figure 2. Bound fraction as a function of bulk concentration whenN= 50 and;ls= 1.0. Solid curve: theoretical calculation; (0): Monte Carlo simulation on a 25 x 25 x 80 lattice; (+I: Monte Carlo simulation on a 50 x 50 x 50 lattice. decrease. In Figure 1,the range used for this purpose is from 8 to 33. With the specification of C#J after the system reaches equilibrium, we can present the properties such as r and 8 as a function of C#J for different N and xs, These results can then be directly compared with the calculations according to the theory of Scheutjens and Fleer. Deviation in Bound Fraction: Figure 2 presents the value of bound fraction calculated from the Monte Carlo simulation (scattered data points) and the calculation results according to the theory of Scheutjens and Fleer (presented as the solid curve) for N = 50 andxs = 1.0.The two points represented by are calculated using a lattice of size 50 x 50 x 50, while the remaining data points are calculated on a lattice of size 25 x 25 x 80. The lattice size effect is small as long as thez direction is much larger than the radius of gyration of the chains. Both models predict a decrease in the bound fraction as the bulk concentration increases. The results from the Monte Carlo simulation lie systematically below the calculation according to the mean-field lattice theory. McCrackin" reported a Monte Carlo simulation of a single chain on a solid surface. The bound fraction for a single chain with N = 50 and segment surface interaction of l.OkBTis about 0.68. This value is lower than the one obtained with the theory near zero concentration. Recently Smith et have performed a Monte Carlo simulation of polymer melts between two parallel surfaces and the results are also compared with the theory of Scheutjens and Fleer as extended to bulk polymer interfaces. They have also reported that the theory predicts a higher fraction ofbeads for adsorbed chains forming trains on the surface than the Monte Carlo simulation. The results from McCrackin, Smith et al. and Figure 2 indicate that an overestimation of the bound fraction by the theory is due to some intrinsic error in the theory. This error is believed to be caused by the allowance of the direct chain back-fold in the theory. The theory counts more conformations available for the chain than it actually has in the Monte Carlo simulation, due to the allowance of the direct back-fold of the chain. That error will give rise to a difference in the probability of the segment distribution in the layers. We will demonstrate this difference by considering the end segment distribution probabilities of a trimer. Let us examine the consequence of assuming the theory has the same value ofpi as in the Monte Carlo simulation. Then the first free segment will distribute among the layers according to these values of pi. If the second
+
Wang and Mattice
2284 Langmuir, Vol. 10,No.7,1994 segment is in layer 2, the statistical weights for the conformations of the first two segments are = (4/6)~2p2; ~; P(1,2) = ( l / 6 ) ~ 9 P(2,2) P(3,2) = (l/6)pg2 (11)
Here P(ij) refers to the probability of the conformation where the first segment is in layer i and second segment is in layerj. This treatment is exact so far. There are a total of six conformations both in the Monte Carlo simulation and in the theory for a dimer. For a trimer, the Monte Carlo simulation has a total of 30 conformations, while the theory counts a total of 36 conformations. According to the theory, the probability that the third segment resides a t layer 1as a result of a random walk of the previous two segments is Ptheory = p1[(1/6)P(1,2)f
......D ......0 .......................
_.". 0
(1/6)P(2,2) + (1/6)P(3,2)1 (12)
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~
...............
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In a nonreversal random walk, there are only five orientations that the third segment can adopt. The probability that the third segment resides in layer 1 according to the Monte Carlo simulation is PMC
=pl[OP(1,2)
x,,, x..........*
+ (1/5)P(2,2) + (1/5)P(3,2)1 (13) 0.14
-
0.04
-
................. ............. x.....................
............. ....."'Y
y.
The difference between the two cases is
6p = 'theory
- P M C =p9&(1/36)p,
+ (-2/9O)P2
-I-
(-1/18O)pJ
(14)
In most situations, p1 > p2,p3 due to the preference for adsorption. Thus 6P is positive. The theory will give a higher probability of the third segment lying on the surface even when the evaluation of pi is correct. So far we have only considered the case where the second segment is in layer 2. If the second segment is in layer 1, the probabilities of all the conformations of the two previous segments are P(1,l) = (4/6)pgl; P(2,l) = ( 1 / 6 ) ~ 9 ~ (15) Then the theory predicts the third segment resides in layer 1 as a result of the above conformations as Ptheory =p1[(4/6)P(l,l)
+ (4/6)P(2,1)1
(16)
Monte Carlo simulation will have
PMC =p1[(3/5)P(l,l) + (4/5)P(2,1)I
(17)
Although the walk toward the 0 layer is forbidden,it should be counted in the total number of conformations.
6p = 'theory
- P M C =p91[(4/9O)p,
+ (-2/90)pJ
(18)
6Pwill be positive even whenpl = p 2 . Thus it is clear that when the back-fold exists, the theory will predict a higher probability that the segment resides near the solid wall due to the error in counting the total number of allowed conformations. The above discussion illustrates the deviation in the probability of the chain segment distribution near the wall. However, such deviation does not exist for the chain segment distribution in a homogeneous bulk phase. If layers 1 and 2 are in the bulk, eq 14 will still apply, but the probability of the free segment in each layer is the same, p 1 = pz = p3. Then 6P in eq 14 is zero. Equations 15-18 will need modification since the zeroth layer is no longer a solid wall. There is another conformation for the two segments noted as (0,l)where the zeroth layer is the
qD.
Q...o.........
............D ................. ................ .a
bulk concentation
Figure 4. Fractional deviation in bound fraction (GPPMc)as N = 5 ; (+) N = a function of bulk concentration. xs = 1.0. (0) 10;(0) N = 30; ( x ) N = 50.
same as the other layers. With this modification, it is again not hard to see 6P will be zero if it is in the bulk phase. Thus the back-fold causes the deviation only when there is a solid wall or a heterogeneous distribution of segments in the layers. Figure 3 presents the difference in bound fraction between the two models as a function ofbulk concentration for different chain lengths. For shorter chains, N = 5 and N = 10, it presents an almost constant deviation at bulk concentrations above 0.1. For longer chains, N = 30 and N = 50, the deviation decreases as the bulk concentration increases and it may reach a plateau at bulk concentrations larger than the ones we have examined here. Figure 4 presents the fractional deviation in bound fraction as a function of bulk concentration. All the data exhibit a constant fraction of overestimation by the theory. The deviation amounts to about 15% when N = 50. The difference in bound fraction can also be caused by the difference in values ofp, evaluated by the theory and by the Monte Carlo simulation. The theory evaluated the value ofpi by using the mean-field approximation in each layer. Thus one cannot exclude the possibility of the influence due to the mean-field approximation at low surface coverage. Here by using the random mixing approximation, the excluded volume interaction of the
Adsorption of Homopolymers on a Solid Surface
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Figure 5. Theoretical surface coverage as a function of bulk concentration, according to the theory of Scheutjens and Fleer using xs = 1.0. From bottom to top: N = 1, 5, 10,30,50,100. segments within the same chain is underestimated by the theory. That would also give rise to an overestimation of the bound fraction by the theory. Note that the bound fraction is an average bound segments per adsorbed chain, and thus the excluded volume interaction of segments within the same chain affects the bound fraction. Such an overestimation should monotonically decrease as the surface coverage increases, and it is most prominent at the lowest surface coverage. The absence of this effect in Figure 4 shows that the deviation is mainly caused by the allowance of the direct chain back-fold for these systems. Deviation in Surface Coverage: Figure 5 presents the theoretical dependence of 8 on the bulk concentration for xs = 1.0 and N from 1 to 100. As N increases, the initial rise of 8 becomes steeper, and in the limit it becomes unmeasurably steep due to the lower limit of concentration one can detect in an experiment or simulate on the lattice. Also note in Figure 5, that the top curve corresponding to N = 100 is almost indistinguishable from the next curve with N = 50. These phenomena are consistent with experimental ~ b s e r v a t i o n . ~ ~Figure - ~ ' 6 presents the 8 calculated from the Monte Carlo simulation (scattered data points) and the calculation for N = 50 according to the theory of Scheutjens and Fleer (presented as the solid curve). The two points represented by are calculated using a lattice of size 50 x 50 x 50 lattice, while the remaining data points are calculated on a lattice of size 25 x 25 x 80. The theory predicts the same qualitative dependenceof 8 on bulk concentration as the Monte Carlo simulation, but there are some quantitative differences. These quantitative differences are appreciable, and in some cases the value of 8 from the simulation exceeds by 0.15 the value from the theory. The lengthy linear extrapolation of the four points with bulk concentration of 0.15-0.30 to a bulk concentration of 1.00 would yield a surface coverage of 0.98, which is close to the expected value of 1. Extrapolation to a bulk concentration of 0 should not be attempted, because the true effect is strongly curved, as shown in Figure 5 . If the theory predicts a higher probability of the sth segment in the first layer, one might expect that the theory will predict a higher 8 value than the Monte Carlo
+
(25)Kawaguchi, M.; Hayakawa, K.; Takahashi, A. Macromolecules 1983,16,631. (26)Kawaguchi, M.; Maeda, K.; Kato, T.; Takahashi, A. Mucromolecules 1984,17,1666. (27)Kawaguchi, M.;Hayashi, K.; Takahashi, A. Macromolecules 1984,17,2066.
Figure6. Surfacecoverage as a function ofbulk concentration for N = 50, xs = 1.0. Solid curve: theoretical calculation; (0) Monte Carlo simulation on a 25 x 25 x 80 lattice; (+) Monte Carlo simulation on a 50 x 50 x 50 lattice. simulation. The results presented in Figure 6 show the contrary. To understand this result, one has to realize that the previous discussion with regard to allowance of direct back-fold of the chain only shows that the theory would give a higher probability that the chain segments reside on the wall if the chains is adsorbed on the surface. 8 is more strongly dependent on the number of chains adsorbed on the surface, although it is also dependent on the bound fraction of the adsorbed chains. In the theory, qNi) and pi are determined self-consistently and pi is directly dependent on Hi)through eq 1. Thus the value of 8 can be underestimated by an underestimation of pi due to the random mixing approximation adopted in determining the value ofpi. Let us assume that the theory accurately predicted the bound fraction for every chain that adsorbed. Then the surface coverage is dependent on the total number of chains adsorbed on the surface. That number is controlled by the excluded volume interaction between the adsorbed chains on the surface. At extremely low surface coverage, such an excluded volume interaction between the chains is small. The random mixing approximationwill also give a very small excluded volume interaction in such cases. Thus no major deviation will be expected. At moderate surface coverage when the chains on the surface are well separated, the excluded volume effect of the adsorbed chains toward each other is actually smaller than using the random mixing approximation since the chain segments are localized. Thus there will be more chains adsorbed on the surface than the theory predicts. When the surface coverage further increases, the deviation will decrease and eventually it will be zero if 4 = 1. It has to be noted here that the affect of the mean-field approximationon the surface coverage is very different than it is on the bound fraction. In the former case it produces the error in evaluating the excluded volume interaction among the adsorbed chains while in the later case it produces the error in the excluded volume interaction of the segments on the same adsorbed chain. The theory assumed a constantpi value in the lateral dimension,but in the Monte Carlo simulation, pi is not constant in the lateral dimension. It is larger inside the adsorbed chain domain, while it is smaller outside the domain of the adsorbed chain. The random mixing would be applicable in two situations. One situation is when the polymer chain is short, when essentially it is a monomeric species. The other
2286 Langmuir, Vol. 10, No. 7, 1994
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Figure 7. Surface coverage as a function of bulk concentration for N = 5, xs = 1.0. Solid curve: theoretical calculation; (0) Monte Carlo simulation. situation is when the local concentration in each layer is high enough that the coils tend to penetrate each other. For the first situation one expects the deviation between the two models would be small when the chain length gets short. Figure 7 presents the comparison of the surface coverage between the two models for N = 5 and xs = 1.0. The deviation is much smaller than in Figure 6, thus indicating that the difference is partly due to the meanfield approximation. The second situation requires that the chains penetrate each other. When the chains are in three-dimensional bulk phases, this penetration will be possible when the concentration is higher than the C*value. However, for the chains that are confined to two dimensions, the chains can not penetrate each other at all.28-30The conformation of the chains at the surface is neither truly twodimensional, nor truly three-dimensional. The stronger the chains are bound to the surface, the closer the chains resemble a two-dimensional chain. Thus one can infer that for the same chain length, when the segments are more strongly attracted to the surface, the chains are more tightly bound to the surface and more deviation will be present. This will be shown to be true later. Figure 8 presents the difference in surface coverage between the theory and Monte Carlo simulation results for several chain lengths. The deviations become more severe as N increases. Also the deviation reaches a maxima at a certain concentration which is dependent on the chain length for N = 5 and N = 10. For the longer chain length, N = 30 and N = 50, we assume the initial rise in the deviation does not show up in the plot due to the limit oflow concentration the Monte Carlo simulation can study. These behaviors are consistent with the previous discussion. Note in Figure 6, the deviation is still quite appreciable when the surface coverage is as high as 0.7. This further indicates the chains have difficulty penetrating one another when they are strongly adsorbed on the surface. It may be noticed that for N = 5 in Figure 8, the first point has a negative value, which means the theory predicts a higher surface coverage than the Monte Carlo simulation results. Later we will present another simulation result which also gives a negative difference in surface coverage. This negative deviation is due to an (28) Bishop, M.; Ceperley, D.; Frisch, H. L.;Kalos, M. H.J.Chem. Phys. 1981,75, 5538. (29)Carmesin, I.; Kremer, K. Macromolecules 1988, 21, 2819. (30) Carmesin, I.; Kremer, K. J. Phys. (France) 1990, 51,915.
Figure 8. Difference in surface coverage (60 = 0MC - etheory) as a function of bulk concentration for xs = 1.0. (+) N = 5; (0) N = 10; (0) N = 30; ( x ) N = 50. 4................--K....................! ...................
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Figure 9. Difference in adsorbed amount (6r= r M C - meolr) as a function of bulk concentration. xs = 1.0. (0)N = 5; (+) N = lO;(U)N=30;(x)N=50. overestimate of the bound fraction for each adsorbed chain by the theory. Its effect only exists a t low surface coverage. Deviation in Adsorbed Amount: Figure 9 presents the difference in absorbed amount, r, for the same systems. The dependence of the deviation of r on the bulk concentration is different from the deviation in surface coverage. For all the systems, the deviation approaches a plateau value at high bulk concentration and it does not decrease to zero. At the limit of 4 = 1, the deviation in 8 is zero while there is a finite difference in bound fraction. Since r, 8, and bound fraction satisfy eq 10,the two models will have a finite deviation in the value of r. The theory of Scheutjens and Fleer would underestimate the adsorbed amount due to the overestimate of the bound fraction. From the above discussion, it is clear that for all the simulations presented so far, both the allowance of direct chain back-fold and the mean-field approximation cause deviations of the theoretical values from the simulation results. Next we will present the simulation with a different surface segment interaction energy and discuss how the deviation between the two models changes as the surface segment interaction energy changes. Effect of xs on the Deviation: Figure 10 presents fractional deviation in bound fraction for both xs = 0.5 and xs = 1.0 for two values of N . The fraction of overestimation of bound fraction by the theory increases
Langmuir, Vol. 10, No. 7, 1994 2287
Adsorption of Homopolymers on a Solid Surface 0.22
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Figure 12. Fractional deviation of surface coverage as a function of surface coverage. (a, 0)x8 = 0.5,N = 10; (a', +) xs = 1.0,N = 10;(b,0 )xs = 0.5,N = 30;(b', x ) x8 = 1.0,N = 30.
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Figure 11. Fractional deviation of bound fraction as a function of surface coverage. (a, 0)x8 = 0.5, N = 10; (a', +) ,ys= 1.0,N = 10; (b, 0 )x8 = 0.5, N = 30; (b', x ) x8 = 1.0,N = 30.
Figure 13. Fractional deviation of adsorbed amount as a function of surface coverage. (a, 0)xs = 0.5, N = 10; (a', +) xs - l.O,N= 10; (b, O)x8= 0 . 5 , N = 30;(b', x ) x s = l.O,N= 30.
as the surface segment interaction energy decreases, although the theory correctly predicts a lower value of the absolute bound fraction with a smaller value of x8. Since the difference in bound fraction can be caused both by the allowance of chain back-fold and mean-field approximation, we replot the data in Figure 10 as a function of surface coverage in Figure 11. For xs = 0.5, the data shows a fast decay as the surface coverage increases. Such a decay indicates that the deviation is caused by the random mixing approximation. The surface coverage has to be higher than the local concentration within the adsorbed chain domain ifthe deviation caused by the random mixing is to disappear. However, apart from that decrease, one can still see that the plateau value of fractional deviation is higher for the smaller value of xs. That plateau value is caused by the allowance of chain back-fold in the theory. This indicates that when the chains adsorbed are not strongly bound to the surface, the allowance of chain backfold will impose a stronger problem. Figure 12 presents the fractional deviation in surface coverage for xs = 0.5 and xs = 1.0. The horizontal axis is the surface coverage. Since the deviation caused by the mean-field approximation is expected to be a function of surface coverage, such a plot will more clearly show the difference between the two situations. With a smaller surface segment interaction, the chains adsorbed on the
surface behave more like in three-dimensions. Thus the deviation caused by the mean-field approximation would be small. Therefore the data with xs = 0.5 lie below the curves for x8 = 1.0in Figure 12. Figure 13is the fractional deviation in adsorbed amount as a function surface coverage. The deviation for x8 = 0.5 is also smaller than for xs = 1.0. The deviation in adsorbed amount is always positive, unlike the deviation in surface coverage which could be negative a t low surface coverage.
Concluding Remarks The comparison reveals some systematic deviations between the theory of Scheutjens and Fleer and the Monte Carlo simulation. The deviations are attributed to two factors: the allowance of the direct back-fold of the chain and the mean-field approximation. The later approximation would work better at weak adsorption, sincethe chains behave more like three-dimensional chains and they can penetrate each other at moderate concentration. However the first approximation works better with stronger adsorption energy, where the chains are more tightly bound to the surface. In the work reported by Smith et al.,24the mean-field approximation does not cause a significant difference since they are studying a polymer melt between two parallel surfaces. The primary deviation between their simulation
2288 Langmuir, Vol. 10,No. 7, 1994
results and the mean-field theory is caused by the allowance of a reversible random walk. Thus they have found quantitative agreement for the chain segment and end segment density distributions between the simulation and the mean-field theory. In our study, both approximations play a role. No quantitative agreement has been found for any of the values monitored in the study. We try to distinguish the source ofthe deviation and degree of that deviation at each particular case. The allowance of direct chain back-fold is closely linked to the fact that the chain is treated as a Markovian chain. Such an approximation will not impose a problem if it is applied in the homogeneous bulk solution. It would give rise to a statistical error when the chain is in inhomogeneous solution or when it encounters a solid impenetrable wall. This is important since the current theoretical models are all based on such an approximation. We have only presented the proof based on a cubic lattice. It is not clear to us how it would behave when it is extended to the continuum limit. We have not studied the effect of segment solvent interaction on these deviations. Some of the probable effectsmay be anticipated based on the knowledge gained
Wang and Mattice from the above discussion. For example, with a bad solvent-segment interaction, the excluded volume effect of the segments within the same chain will decrease. It can behave like an ideal chain. Under such situation, the effect of the mean-field approximation on the bound fraction may disappear. However, the allowance of direct chain back-fold will still cause a deviation. This deviation can be removed with a more sophisticated theory where the knowledge of three successive segments is necessary. Leermakers, Scheutjens, and Gaylord3I have given such an approach which eliminates the direct back-fold of the chain. A comparison of such calculations with our Monte Carlo simulation would be interesting. It would also further clarify the validity of the assumption of random mixing adopted in the theory.
Acknowledgment. This research was supported by a gift from the BFGoodrich Co. The calculation according to the theory of Scheutjens and Fleer is done using Simpolsoft software Polad version 9.01. (31)Leermakers, F.A. M.; Scheutjens, J. M. H. M.; Gaylord, R. J. Polymer 1984,25, 1577.
