pubs.acs.org/Langmuir © 2009 American Chemical Society
Orientational Order in Nanolayers of Cast Polymer Films A. V. Maksimov,*,† G. M. Pavlov,*,‡ and I. V. Kusheva† †
Cherepovets State University, Lunacharskii Pr. 5, Cherepovets, Vologda region, 162600 Russian Federation, and ‡Institute of Physics, St. Petersburg University, Ulianovskaya Street 1, 198504 St. Petersburg, Russian Federation Received February 11, 2009. Revised Manuscript Received July 6, 2009
The spontaneous ordering of fragments of chain molecules near the surface in polymer films is described in terms of the multichain model, which allows for local intra- and interchain segment orientational interactions, as well as for transversal fluctuation of their orientation as an approximation of strong planar orientation order in the layers. Chain packing in the plane-ordered state is impossible unless the interchain interaction parameter has a critical value. This value decreases with chain bending rigidity. The calculated dependences of the limiting values of the quadrupole orientation order parameter on the length of the Kuhn statistical segment describe reasonably well the experimental data obtained in a study of polymer homologue polysaccharides and sulfonated phenyl-containing polymers. The monolayer thickness in films of some polysaccharides and sulfonated phenyl-containing polymers has been calculated from the fit of the theoretical to the experimental data of the surface birefringence on the number of nanolayers.
1. Introduction The study of physicochemical characteristics of polymer films and their near-surface layers is of considerable importance for practical applications, as well as for fundamental polymer science. In modern applications, thin polymeric films serve as protective surface coatings, adhesives, membranes, lithographic materials, etc.1-3 Polymer films exhibit properties other than those of bulk polymers and, as such, have become a subject of extensive theoretical and experimental research.1-6 One of the central ideas that guide the investigation and interpretation of the properties of polymer films is the assumption that fragments of polymer molecules are more ordered near the surface. It is possible to outline three general thickness categories of structured polymeric films: monolayer Langmuir-Blodgett films, smectic multilayer films, and three-dimensional ordered layered structures. Films in polymer science include concepts from both Langmuir-Blodgett monolayers and multilayered structures, up to hundreds of micrometers in thickness.1-3 Three-dimensional multilayers composed of several monolayers (on the order of 10), capable of forming smectic structures under certain conditions, occupy an intermediate place in terms of thickness. Threedimensional ordered layered structures are also observed in biomembranes, n-paraffins, and polyurethanes.7-9 For example, in concentrated solutions and melts of comb-shaped polymers *To whom correspondence should be addressed. E-mail addresses:
[email protected];
[email protected]. (1) Physics of Polymer Surfaces and Interfaces; Sanchez, I.C., Fitzpatrick, L.E., Eds.; Butterworth-Heinemann: Boston, 1992. (2) Polymer Surfaces, Interfaces and Thin Films; Karim, A., Kumar, S., Eds. World Scientific: Singapore, 2000. (3) Mulder, M. Basic Principles of Membrane Technology; Kluwer: Dordrecht, The Netherlands, 1996. (4) Mischief, C.; Baschnagel, J.; Dasgupta, S.; Binder, K. Polymer 2002, 43, 467. (5) Saulnier, F.; Raphael, E.; De Gennes, P.-G. Phys. Rev. E 2002, 66, 061607. (6) Chen, N.; Maeda, N.; Tirrell, M.; Israelachvili, J. Macromolecules 2005, 38, 3491. (7) Magonov, S.; Yerina, N; Ungar, G.; Reneker, D. H.; Ivanov, D. A. Macromolecules 2003, 36, 5637. (8) Liquid Crystalline Order in Polymers; Blumstein, A., Ed. Academic Press: New York, 1978. (9) Shibaev, V. P.; Moiseenko, V. M.; Freidzon, Ya. S.; Plate, N. A. Eur. Polym. J. 1980, 16, 277.
Langmuir 2009, 25(16), 9085–9093
that form supermolecular smectic LC structures,9 the main chains are arranged in the plane and linear side fragments are oriented normal to the plane. Such fragments can enter into the ordered quasi-lattice practically without distortion in some cases.10 There exist a variety of polymer systems that comprise quasitwo-dimensional (layered) nanostructures in which interactions between elements in the same layer are substantially stronger than those between elements from different layers. Such systems are exemplified by polyimides, a class of polymers with a notably high heat resistance. X-ray studies of some polyimide types show that the main chains of macromolecules in crystalline regions are packed in an almost parallel manner. Pyromellitic imides and phenyl rings from different macromolecules form “parquet-like” tilted layers or another kind of packing inside the layer.11 Conformational calculations lead to the conclusion that rings belonging to one layer weakly interact with rings from the neighboring layers and that the layer structure and the retarded rotation of rings are primarily determined by the interactions of cyclic groups inside one layer.12 Another set of examples are polycarbonates in the glassy state. It is supposed that layered domains with a relatively regular quasi-crystalline packing exist in these polymers, in which parts of chains are arranged parallel to one another and major obstacles exist to the rotation of the phenyl ring along the C-C bond due to interactions with one another in the plane perpendicular to the orientation axis of molecular backbones.13 Such systems are characterized by the parallel packing of long axes of molecules, while short axes either are randomly oriented in the plane perpendicular to the long axis or form an ordered structure in which the order-disorder transition is possible. Therefore, the structure and molecular dynamics of such systems can be described in terms of the pure (10) Shalyganova, Yu. E.; Kerber, M. L.; Chalykh, A. E. In Proceedings of IV All-Russia Conference “Structure and Dynamics of Molecular Systems”; Kazan. Gos. Univ.: Kazan, Russian Federation, 1997; Vol. 1, p 21. (11) Baklagina, Yu. G.; Milevskaya, I. S.; Efanova, N. V..; et al. Vysokomol. Soedin., Ser. A 1976, 18, 1235. (12) Zubkov, V. A.; Milevskaya, I. S.; Baklagina, Yu.G. Vysokomol. Soedin., Ser. A 1985, 27, 1543. (13) Kolynsky, A.; Skolnick, J.; Yaris, R. Macromolecules 1986, 19, 2550.
Published on Web 07/21/2009
DOI: 10.1021/la9005168
9085
Article
two-dimensional model14-16 or the three-dimensional planarchain model.17-19 The long-range orientational order in ordered polymer chains may be characterized accordingly by dipole or quadrupole orientational order parameters P0 Æcos γnæ and S0 1/2[3Æcos2 γnæ - 1] (γn is the angle between the nth segment axis and the axis of order, the director in liquid crystalline systems,8 or the normal to the film (substrate) surface in the case of polymer films). Parameter P0 = S0 = 0 in the isotropic state with random orientation of chain segments and P0 f 1 and S0 f 1 in strongly axially ordered systems, for example, in ferroelectric and nematic phases accordingly. In polymer films when S0 = -0.5, the perfect planar-chain ordered system is observed.20-25 In general, when the range of angle γn is 125.3 > γn > 54.7 the chains are oriented predominantly in the plane of film surface; when 54.7 > γn > -54.7, the chains are largely arranged normally to the surface.23 It is necessary to mention that the orientational order parameter P0 determines dielectric properties of chains with oriented dipole moments directed along the chain segments in polar polymer systems, whereas the relaxation of quadrupole orientational order parameter S0 determines the luminescence depolarization for a chain containing a marker with an oscillator directed along the axis of the chain segment. This parameter may be also related to nuclear magnetic relaxation or measurement of spontaneous birefringence.26 An informative method for studying the ordering of fragments of macromolecules in films and membranes is the tilted polarized beam method.20,21 This is one of the effective tools for studying polymer films based on the measurement of spontaneous birefringence that emerges in polymer films. It has been found that the effect of spontaneous surface birefringence is observed for films prepared simply from solutions through free evaporation of solvent, without application of any additional physical fields. Due to the different interactions with the support of the polymersolvent system, the value and sign of surface birefringence vary depending on the chemical structure of the polymer repeating unit and the character of orientation of anisodiametric macromolecular fragments near the surface. The Cherkasov-Vitovskaya-Bushin phenomenological theory makes it possible to separate the structural and orientational contributions to spontaneous surface birefringence.20 Later, the layer model of films was proposed, which explains the effect of saturation of spontaneous birefringence with an increase in the film thickness.22 With the use of a series of polysaccharides and sulfonated phenyl-containing polymers, it was shown that the (14) Luckhurst, G.; Simpson, P.; Zannoni, C. Liq. Cryst. 1987, 2(13), 13. (15) Gotlib, Yu.Ya.; Darinskii, A.A. ; Lyulin, A. V.; Neelov, I. M. Vysokomol. Soedin. 1990, A32, 810. (16) Maksimov, A.V.; Maksimova, O.G. Polym. Sci. A 2003, 45, 1476; Vysokomol. Soedin., Ser. A 2003, 45, 1476. (17) Maksimov, A. V.; Gotlib, Yu. Ya. Vysokomol. Soedin., Ser. A 1988, 30, 1411. (18) Gotlib, Yu.Ya.; Maksimov, A. V. Vysokomol. Soedin., Ser. A 1988, 30, 2561. (19) Gotlib, Yu. Ya.; Maksimov, A. V. Vysokomol. Soedin., Ser. A 1990, 32, 1455. (20) Cherkasov, A. N.; Vitovskaya, M. G.; Bushin, S. V. Vysokomol. Soedin. 1976, 18, 1628. (21) Grishchenko, A. E.; Cherkasov, A. N. Usp. Fiz. Nauk. 1997, 167, 269. (22) Grishchenko, A. E.; Pavlov, G. M.; Vikhoreva, G.A. Polym. Sci. B 1999, 41, 1347; Vysokomol. Soedin., Ser. B 1999 41, 1347. (23) Pavlov, G. M.; Grishchenko, A. E. Biotechnol. Genet. Eng. Rev. 1999, 16, 347. (24) Pavlov, G. M.; Grishchenko, A.E. Polym. Sci. B 2005, 47, 1882; Vysokomol. Soedin., Ser. B 2005 47, 1882. (25) Pavlov, G. M.; Grishchenko, A. E.; Ryumtsev, E. I..; et al. Biofizika 1999, 44, 251. (26) Gotlib, Yu. Ya., Darinskii, A. A.; Svetlov, Yu. E. Physical Kinetics of Macromolecules; Khimiya: Leninigrad, 1986.
9086 DOI: 10.1021/la9005168
Maksimov et al.
Figure 1. Structural formulas of repeating units of some polysaccharides and phenyl-containing sulfonated polymers.
degree of orientational order of anisodiametric fragments of macromolecules is unambiguously related to structural-conformational characteristics of macromolecules, such as the statistical segment length (Kuhn segment) or the persistence length. Experimental values of the quadrupole orientational order parameter in surface layers for the polymers in question are negative;23-25 this implies that fragments of chains form planar orientational order in surface layers; that is, aligned primarily along the film surface. The planar-chain model is useful for the description of the orientationally ordered state of rigid-chain polymers solely in which chains are arranged virtually in the planes.17-19 At the same time, fragments of chains can occur outside these planes in the surface layers of polymers that exhibit equilibrium flexibility. The objective of this study was to theoretically examine the dependence of the orientational order parameter on the rigidity of polymer chains, as well as the dependence of the surface birefringence on the thickness for some polymer-homologue polysaccharides and phenyl-containing sulfonated polymers (see Figure 1). For this purpose, we considered a model in which chain segments can deviate from the direction of the planar orientational order. Langmuir 2009, 25(16), 9085–9093
Maksimov et al.
Article
The fluctuations of orientation of a chain segment (unit vector dn) at any quasi-lattice site n relative to the preferable orientation of chain segments (x-axis in Figure 3) can be described with the use of three components, {δdn}x, {δdn}y, and {δdn}z, the projections of the vector δdn = d0 - dn. In these coordinates, the potential of interaction (defined by eq 1) between two segments that occur at sites n and m will take the form27,28 VnðiÞ, m ¼ Ki ½fδdn gx þ fδdm gx -ðδdn , δdm Þ,
Figure 2. Multichain model of a three-dimensional ordered system consisting of N3 layers. See the text for detail.
2. Modeling In the planar-chain model, it is supposed that N2N3 polymer chains (each composed of N1 rigid segments with a length l smaller than the statistical segment) occur completely in-plane and form a three-dimensional ordered quasi-lattice (in Figure 2, N2 and N3 are the number of chains along each direction of the cross section of the system of chains).17-19 The position of a rigid chain segment in the quasi-lattice is determined by the set of three numbers (n1, n2, n3) n. The quantity n1 is counted along the contour length N1l of the given chain (n1 = 1, ..., N1); n2 and n3 are the numbers of the neighboring segments of different chains (n2 = 1, ..., N2 and n3 = 1, ..., N3). The state of the strong planar orientation order in layers is described by the unit vector d0 (which corresponds to the direction of the director in a liquid crystal). The angle j characterizes the orientation of chain segments with respect to d0 (Figure 2). In the planar-chain model,17-19 the energy constant K1 describes intrachain orientational interactions and is related to the thermodynamic bending rigidity and K2 describes interchain orientational interactions in layers and between the layers. Let us consider a slightly different model in which the orientation of chain segments may deviate near d0 (Figure 3). This model (i) between two suggests that the potential energy of interaction Vn,m chain segments that occur at quasi-lattice sites n and m depends on the angle Φn,m between their axes in the (x, y, z) space VnðiÞ, m ¼ -Ki cos Φn, m
ð1Þ
and not on the angle j of their mutual orientation in the layer plane (x, y) as in the planar-chain model.17-19 In spherical coordinates (Figure 3), the cosine of this angle may be expressed in terms of corresponding polar and azimuthal angles: cos Φn,m = cos θn cos θm + sin θn sin θm cos(jn - jm). In eq 1, the superscript i is equal to unity for segments belonging to one chain or i = 2 for segments from different chains. The chain bending rigidity is determined by the constant ~ = K1, which defines the number of repeating units A A/l ~ 4K1/(kBT) in the Kuhn statistical segment A (or reduced persistence length a = A/2; see Table 1, columns two and three). This model is an averaged model in which possible irregular arrangement of chains with respect to a given chain in actual ordered systems is replaced by regular arrangement characterized by the effective interchain interaction constant K2 or by the dimensionless parameter b = 2K2/(kBT) analogous to a. In the general case, it is assumed that K1 > K2; that is, the preferable action of orientational forces along the axes of macromolecules is admitted. The constants K1 and K2 for polar and nonpolar chains were evaluated in ref 16. Langmuir 2009, 25(16), 9085–9093
i ¼ 1, 2
ð2Þ
Let us introduce spherical polar coordinates in which the x-axis is parallel to the vector d0; that is, the d0 = (1, 0, 0). In these coordinates, the orientation of a chain segment is characterized by two angular variables, jn and θn (Figure 3). In the approximation of a strong order in layers (θn , π), the fluctuations of the vector projections δdn on the z- and y-axes for chain segments are {δdn}z = cos jn sin θn ≈ θn and {δdn}y ≈ sin jn sin θn ≈ θn, whereas the deviations of the projections on the x-axis are {δdn}x = 1 - cos θn ≈ θn2/2, that is are of the second-order of magnitude in θn. At small angles θn, the first two linear terms in potential eq 2 are of the same order of magnitude as the quadratic terms (the exception is the term (δdn, dm), which has the next order of magnitude). The components of the vector δdn are related by the expression 2fδdn gx ¼ fδdn gy 2 þ fδdn gz 2
ð3Þ
This expression is derived from the condition that the normalization of the segment orientation vector dn is retained upon statistical “distortions”29 in the given approximation (secondorder magnitude in θn): (d0 - δdn)2 = (dn)2 = 1. Equation 3 allows the longitudinal (with respect to the direction of x) variables {δdn}x to be ignored and leaves only the transverse components {δdn}z and {δdn}y statistically independent. Therefore, in the vicinity of the ordered state, the effective potential energy of interaction of the neighboring chain segments has the quadratic form in the variables {δdn}R (R = z, y) ( X 1 X K1 ½fδdn gR -fδdm gR 2 Vef ½fδdn gR ¼ 2 R ¼z, y jn1 -m1 j ¼1 ) X X 2 þK2 ½fδdn gR -fδdm gR ð4Þ jn2 -m2 j ¼1 jn3 -m3 j ¼1
3. Statistical Properties of Infinitely Extended Systems Using the normal-modes analysis,29,30 we can calculate different equilibrium averages Æ{δdn}R{δdm}Ræ to characterize the correlation between the fluctuations of projections of segments that occur at two different quasi-lattice sites n and m (R = z, y) and the mean-square projections of the vector h on the z-, y-, and x-axes. The calculation of these means via the normal-modes analysis31 for the components of the vector δdn leads to expressions for correlation between the fluctuations of projections of segments and mean-square projections of the end-to-end chain vector h similar to those obtained by de Gennes32 for the (27) Gotlib, Yu.Ya.; Maksimov, A. V. Vysokomol. Soedin., Ser. B 1987, 29, 822. (28) Gotlib, Yu.Ya.; Baranov, V. G.; Maksimov, A. V. Vysokomol. Soedin., Ser. A 1987, 29, 2620. (29) de Gennes, P.-G. The Physics of Liquid Crystals; Clarendon: Oxford, U.K., 1974. (30) Landau, L. D.; Lifshitz, E. M. Statistical Physics; Nauka: Moscow, 1976; Pergamon: Oxford, U.K., 1980. (31) Maradudin, A.; Montroll, E.; Weiss, G. H. Theory of Lattice Dynamics in the Harmonic Approximation; Academic: New York, 1963; Mir: Moscow, 1965. (32) de Gennes, P.-G.In Polymer Liquid Crystals; Ciferri, A., Krigbaum, W. R., Meyer, R. B., Eds.; Academic Press: New York, 1982.
DOI: 10.1021/la9005168
9087
Article
Maksimov et al.
Figure 3. Fluctuations of the axis orientation vector dn for the given chain segment at a site n about the direction of the planar long-range orientation order (director) d0 in the Cartesian (x, y, z) and the spherical polar (θ, j) coordinates.
Table 1. Experimental Value of Equilibrium Rigidity and Surface Birefringence Parameters for Some Polysaccharides and Phenyl-Containing Sulfonated Polymers from refs 23 and 24 length A~ of statistical Kuhn segment, nm
polymer
number of repeated units in ~ statistical segment, A A/l
limiting value of orientational order parameter S0
ΔB/ΔH, s m-1
limiting value of surface birefringence B0
dextran 1.3 2.3 -0.007 0.72 0.021 16 31 -0.29 30 0.212 HPMCa 48 0.191 chitosan 21 41 (-0.36)b a b 24 47 (-0.35) 0.083 CMCh sodium 4 8.4 -(0.14-0.085) -(49 ( 3) -(0.798-0.485)b polystyrene-4sulfonate SAPA-meta 9 18.1 -0.19 +(210 ( 10) -(0.532)b SAPA-para 70 143 (-0.39)b +(770 ( 50) (0.016)b a HPMC, hydroxypropyl methyl-cellulose; CMCh, carboxymethyl chitin. b The values received by means of theoretical calculations.
continuum model of nematic single crystals (see detailed analysis in Appendix A). The quadrupole orientational order parameter of chain segments is defined by the expression S0 1/2[3Æcos2 γnæ - 1 = 1 /2[3Æθn2æ - 1] (γn is the angle between the segment axis and the normal to the film surface, Figure 3). By means of the normalmodes analysis used in the Born-von Karman theory of lattice dynamics31 in the harmonic approximation, S0 is reduced to the form 1 3 ð5Þ S0 ¼ ½ IðβÞ -1 2 b where β = a/b and I( β) is the triple integral defined as IðβÞ ¼
1 π3
Z
π 0
Z
π 0
Z
π 0
dψ1 dψ2 dψ3 ð6Þ 2 þ β -cos ψ3 -cos ψ2 -β cos ψ1
Equation 5 can be represented in the equivalent form 1 S0 ¼ ½ðbc =bÞ -1 2
ð7Þ
Here, bc is the critical value of the interchain interaction parameter (b) at which the quadrupole orientational order parameter S0 is zero, that is, the transition to the isotropic phase takes place. A comparison of eqs 5-7 leads to the conclusion that, at a given equilibrium rigidity (parameter a), the value 9088 DOI: 10.1021/la9005168
of bc is determined from the solution for the transcendental equation bc ¼ 3Iða=bc Þ
ð8Þ
Let us assume that the magnitude of interchain interactions in the solvent evaporation process during the formation of polymer films mainly depends on the concentration of chains in the system. Such a situation is observed in athermal solutions of noninterpenetrating rod macromolecules, that is, in lyotropic liquid crystals.29 Then the critical-point condition given by eq 8 is attained at a certain concentration when b = bc and a second-order phase transition from the isotropic (S0 = 0) to the plane-ordered state (-0.5 e S0 < 0 at b > bc) takes place. The value of b = ¥ in eq 7 corresponds to the ideal planar packing (S0 = -0.5). The dependence of bc on the chain bending rigidity (parameter a) is analyzed in detail in Appendix B. The possibility of existence of a nematic or an axially ordered state is not considered in this model, although eq 7 formally suggests that 0 < S0 e 1 at bc/3 e b < bc. In this interval of values of b, the model is not valid anymore, since the transversal fluctuations of orientation of segments will be considerable under these conditions. In this case, one should employ other models33,34 that allow thermotropic transitions to the nematic or (33) Medvedev, G. A.; Gotlib, Yu. Ya. Vysokomol. Soedin., Ser. A 1991, 33, 715. (34) Maksimov, A. V. Polym. Sci. A 2008, 50, 518; Vysokomol. Soedin., Ser. A 2008, 50, 518.
Langmuir 2009, 25(16), 9085–9093
Maksimov et al.
Article
Figure 4. The double logarithmic dependence of the absolute value of the quadrupole orientational order parameter S0 on the reduced length A = 4K1/(kBT ) of a Kuhn statistical segment. Experimental data of spontaneous birefringence in polysaccharide films (O) 24 and phenylcontaining sulfonated polymers (/)34 and theoretical calculations at b = 1.75 (1) for polysaccharide films35,36 and b = 1.1 (2) for phenylcontaining sulfonated polymers.36 Dotted parts of dependences 1 and 2 correspond to the range of A values when the model becomes invalid. Table 2. Calculated Values of Parameters Characterizing the Films of Some Polysaccharides and Phenyl-Containing Sulfonated Polymers
polymer dextran HPMC chitosan CMCh sodium polystyrene-4-sulfonate SAPA-meta SAPA-para
critical interchain interaction parameter, bc
interchain interaction parameter, b = 2K2/(kBT)
effective film thickness, H0, μm
monolayer thickness, d, nm
number of layers, N0 = H0/d
1.57 0.24 0.16 0.18 0.78 0.40 0.05
1.76 1.28 2.21 1.75 1.23 0.99 1.08
178 49 35 119 130 20 0.17
185 15 11 10 400 300 33
960 3270 3180 11900 325 67 5
smectic stable state in which the chains can pack normal to the surface layer. The dependences of the value of the quadrupole orientational order parameter S0 on the normalized (reduced) length of the statistical Kuhn segment A = 4K1/(kBT) (in amount of repeating units in the chain) for polysaccharides and phenyl-containing sulfonated polymers are shown in Figure 4 in the log-log representation. Different polymers in this model are described by different values of the local interchain interaction parameter b = 2K2/(kBT) corresponding to the chain rigidity parameter A and the order parameter S0 (Table 1, columns 2-4, and Table 2, column 3). However, as is seen from Figure 4, the best fit to the experimental data23,24 is reached with the function S0(A) calculated according to eq 5 at the same value of the interchain interaction parameter, b = 1.75 for different polysaccharide structures and b = 1.1 for phenyl-containing sulfonated polymers.35-37 Thus, on the one hand, different chemical microstructure and orientations of monomers (Figure 1) lead to the formation of chains with substantially different equilibrium characteristics. On the other hand, interchain interactions in the given polysaccharides and phenyl-containing sulfonated polymers can be described in terms of this model with the same mean value of the parameter b. For chains with a higher rigidity (A . 1), the order parameter S0 tends to -0.50 (Figure 4), a value that is fully consistent with the plane chain conformations as in the planar-chain model.17-19 This state may be realized for the rigid chain polymers: for SAPApara (A = 70 nm) from phenyl-containing sulfonated polymers and for xanthan (A = 240 nm) from polysaccharides due to chemical structure of the chains (Figure 1). Langmuir 2009, 25(16), 9085–9093
4. Finite Nanolayers of Polysaccharides and Sulfonated Phenyl-Containing Polymers During the calculation of the quadrupole orientational order parameter for a system (Figure 2) consisting of a finite number N3 of extended layers (N1, N2 . 1), integration in eq 6 by means of the normal-modes analysis31 over the variable ψ3 in direction 3 perpendicular to the layers should be replaced with summation, and the order parameter Z Z 3 1 X π π dψ1 dψ2 1 S ¼ 2bπ2 N3 ψ 2 2 þ β -cos ψ -cos ψ -β cos ψ 0 0 3, n 2 1 3, n
ð9Þ The values of the components ψ3,n of the wave vector in sum 9 for a finite system (where n = 1, 2, ..., N3) depend on the type of boundary conditions imposed on fluctuations of transversal projections {δdn}R of chain segments (R = z, y) that occur in the extreme layer planes (n = 0 and n = N3 + 1, Figure 2). (1)
Periodic conditionsðfδdn1 , n2 , 0 gR ¼ fδdn1 , n2 , N3 gR Þ; ψ3, n ¼ 2πðn -1Þ=N3
(2)
Free conditions ðfδdn1 , n2 , 1 gR ¼ fδdn1 , n2 , 0 gR , fδdn1 , n2 , N3 þ 1gR ¼ fδdn1 , n2 , N3 gR Þ; ψ3, n ¼ πðn -1Þ=N3
(3)
Fixed conditions ðfδdn1 , n2 , 0 gR ¼ fδdn1 , n2 , N3 þ 1gR ¼ 0Þ; ψ3, n ¼ πn=ðN3 þ 1Þ DOI: 10.1021/la9005168
9089
Article
Maksimov et al.
Figure 5. Quadrupole orientational order parameter S as a function of the number of layers N3 for (1-4) chitosan and (5-8) dextran under (1, 5) free, (2, 6) periodic, (3, 7) fixed, and (4, 8) semifree boundary conditions. S01 and S02 are the corresponding values of the order parameter in the saturation region. The interchain interaction parameter b = 2K2/(kBT) is 1.75.
(4)
Semifree conditions ðfδdn1 , n2 , N3 þ1 gR ¼ fδdn1 , n2 , N3 gR fδdn1 , n2 , 0 gR Þ ¼ 0; ψ3, n ¼ πð2n þ 1Þ=ð2N3 þ 1Þ
Calculations have shown that the quadrupole orientational order parameter S increases with a growth in the number of layers N3 for the fixed and semifree boundary conditions and, conversely, decreases for the periodic or free boundary conditions (Figure 5). However, regardless of the type of boundary conditions, as N3 increases, S tends to the same limiting value S0 defined by eq 5. In this context, to obtain quantitative fitting with the experimental plots of the surface birefringence coefficient B(H ) for calculation of the theoretical dependence of the spontaneous birefringence on the number of layers N3 in terms of the model proposed in this study, the free boundary conditions were selected and the corresponding equation BðN3 Þ ¼ -cSðN3 Þ
thickness H for (1) chitosan, (2) carboxymethylchitin, and (3) dextran films. The solid lines represent the results of theoretical calculation. The circles, triangles, and squares are the experimental points of the measurements of the spontaneous birefringence, the dotted lines show the initial slope (dB/dH) of B(H) functions,35 and the dot-dash curves are the extrapolations of experimental data.24
elongation (“planarity”) of its chains, so much smaller thickness of one monolayer. Value of d makes it possible to determine the number of layers N0 = H0/d corresponding to the effective thickness H0 of anisotropic surface layers (Table 2, columns 4-6). Let us define the limits of application of the model to the description of orientational order in flexible-chain polymers in which self-assembling effects are less pronounced than in rigidchain polymers. Since the orientational order parameter in eq 7 is S0 e 1, the interchain interaction parameter is b g bc. In terms of the model in question, the relationship b = 1.75 > bc = 1.57 holds for the most flexible chains (dextran) in this polysaccharide series (Table 2, columns 2-3). Consequently, the model employed in this study is applicable to all of the structures compared. If the interchain interaction parameter had a somewhat lower value (e.g., b = 1.50), the calculated function S0(A) would not explain the experimental value of S0 for dextran (cf. the dashed parts of curves 1 and 2 in Figure 4).
ð10Þ
was used, where the fitting coefficient c = B0/(-S0) is the ratio of limiting values of variables B and S (at N3 f ¥), correspondingly. More detailed determination of this coefficient from measurements of spontaneous surface birefringence is explained in Appendix C. The dependences S(N3) were calculated from eq 9 with the determined above values of interchain interaction parameters b for different polysaccharide structures and phenyl-containing sulfonated polymers (Table 2, column 3) and with the proviso that N3 = H/d (d is the monolayer thickness). As is seen from Figures 6 and 7, dependences B(H ) describe in the best manner the experimental data on the spontaneous birefringence for chitosan, carboxymethylchitin, and dextran films at a monolayer thickness d. Scale of this thickness has an order of some nanometers (Table 2, column 5). A smaller last value of d for the rigid-chain polymers and more large interchain interactions may be explained by the greatest Kuhn segment that promotes a greater degree of (35) Pavlov, G. M.; Gubarev, A. S.; Zaytseva, I. I.; Fedotov, U. A. Polym. Sci. A 2007, 49, 1476; Vysokomol. Soedin., Ser. A 2007, 49, 1476. (36) Maximov, A. V.; Pavlov, G. M. Polym. Sci. A 2007, 49, 1239; Vysokomol. Soedin., Ser. A 2007, 49, 1239.
9090 DOI: 10.1021/la9005168
Figure 6. Surface birefringence coefficient B as a function of
5. Summary The behavior of polymers in films or close to interfaces is far from being understood. Many observations, encompassing both structural and dynamical behavior, indicate that the properties of polymers in films deviate from what we know from the bulk. This is due, in part, to the increasing influence of entropic effects (confinement and chain fragment orientation or packing). The interfacial sensitivity highlights the importance of the properties of the near surface region in polymer films; a topic whose importance is beginning to be recognized. The goal of this study is the development of a theoretical multichain model describing the spontaneous ordering of macromolecules in nanolayers near the surface. The model of the polymer chain near the surface, which takes into account the local intra- (K1) and interchain (K2) orientational interactions, as well as transversal fluctuation of orientations near the fully planar ordering is proposed. Analytical calculations developed for the proposed model are consistent with the experimental results captured for the film birefringence of different polymers. The theoretical dependence of the limiting value of the orientational quadrupole order parameter S0 vs the normalized statistical Langmuir 2009, 25(16), 9085–9093
Maksimov et al.
Article
Figure 8. The critical parameter of interchain interactions bc versus the reduced chain rigidity (a = 2K1/(kBT)) as obtained from (1) numerical solution of exact eq 8 and in the (2) rigid-chain (β .1) and (3) flexible-chain (β , 1) approximations. Point A corresponds to the critical value of the interchain interaction parameter of b c = 1.52 for systems with isotropic interactions (β = 1).
by means of the tilted polarized-beam technique and contributes to the fundamental understanding of the macromolecular selforganization near the surface. This model is consistent with the prediction of the phenomenological layer theory used for the interpretation of spontaneous birefringence effect20-25 in polymer films and finally allows us to estimate the thickness of one film monolayer.
Figure 7. Coefficient of the surface birefringence B as a function of the film thickness H: (a) for SAPA-meta (1) and sodium polystyrene-4-sulfonate (2); (b) for SAPA-para. The triangles, squares, and circles are the experimental points of the measurements of the spontaneous birefringence, the dot-dash lines show the initial slope (dB/dH) of B(H) functions,35 and the solid lines represent the results of theoretical calculation.37
segment length A = 4K1/(kBT) is found. The choice of corresponding values of the interchain interaction parameter b = 2K2/(kBT) allows one to describe satisfactorily the experimental results obtained for polysaccharide films and films obtained from sulfated phenyl-containing polymers. The surface birefringence proves that the orientation of polymer chain fragments near surfaces in most cases is predominantly planar. Such orientation ordering is possible only when the parameter b > bc. The critical value bc decreases with increasing chain rigidity (A). Ideal planar order of chain segments (S0 f -0.50) may be realized for the most rigid polymers like to xanthan (A = 240 nm). Experimentally observed linear dependence of the surface birefringence coefficient B vs the polymer thickness H (at low H ) corresponds to the periodic and free boundary conditions in the considered model. For higher values of thickness, the saturation of the dependence B(H ) may be observed with the limiting value of B0. The thickness d of one film monolayer may be estimated by the superposition of theoretical and experimental dependencies of the orientational quadrupole order parameter S(A) and the birefringence coefficient B(H ). The proposed model adequately describes the experimental data obtained via the measurement of spontaneous birefringence Langmuir 2009, 25(16), 9085–9093
Acknowledgment. The authors thank I. I. Zaitseva and A. S. Gubarev for their help in the SB measurements. The authors are also grateful to E. F. Panarin and Yu. A. Fedotov for providing of water-soluble phenyl-containing polymers. The authors would like also to acknowledge Joseph T. Delaney, Jr., for his help in proofreading of the manuscript. This work was supported by the Federal Target Program “Scientific and ScientificallyPedagogical Manpower of Innovational Russia on 2009-2013” (code 263-P).
Appendix A: Correlation between the Fluctuations of Projections of Segments and Mean-Square Projections of the End-to-End Chain Vector h An expression for the longitudinal components of orientation fluctuations of axes of segments that occur at a distance r = (r1, r2, r3) from one another along the corresponding directions of the quasi-lattice n (Figure 2) is Gxn þr, n ¼ Æfδdn gx fδdn þr gx æ=0
ðA1Þ
Equation A1 holds only at the second-order magnitude of this approximation in θn in which the quadratic fluctuations of orientation of segments along the x-axis may be neglected. For the infinitely extended system (N1, N2, N3 > 1), the correlations of fluctuations G(R) n+r,n= Æ{δdn }R{δdn+r}Ræ (R = z, y) do not depend on the position of segments in chains because of spatial uniformity, being determined only by the relative arrangement of the segments in the quasi-lattice n (Figure 2), that is, Gxn+r,n G(R) r By means of the standard normal-modes analysis, which is used to calculate the Green function for a cubic crystal lattice [ref 31, pp 200-201] or orientation fluctuations in the continuum model of low-molecular-mass nematic single crystals DOI: 10.1021/la9005168
9091
Article
Maksimov et al.
a [ref 29, pp 122-124], the correlation function G(R) r G (r1, r2, r3) is reduced to the form
GðRÞ ðr1 , r2 , r3 Þ Z πZ πZ π 3 cosðr1 ψ1 Þ cosðr2 ψ2 Þ cosðr3 ψ3 Þ dψ1 dψ2 dψ3 ¼ 2bπ2 0 0 0 2þβ -cos ψ2 -cos ψ3 -β cos ψ1
ðA2Þ In eq A2, the parameter β = a/b = K1/K2 characterizes the anisotropy of intra- and interchain interactions. Correlation between the orientations of chain segments in the layer planes decreases according to the following law with an increase in the distance (number of segments) r ¼ ðr3 2 þr2 2 þðr1 2 =βÞÞ1=2 :29,31 GðRÞ r =γ=ð2rÞ 1=2
ðA3Þ 1=2 -1
where γ ¼ kB T=ð2πðK1 K2 Þ Þ ¼ ðπðabÞ Þ A (1/r)-decrease in the transversal correlation was found in all physical systems in which interactions are governed by shortrange forces,29 and the ordered state is characterized by the dedicated direction d0. This direction in the given case is arbitrary (degenerated) in the layers’ plane. For comparison, it should be noted that the corresponding correlation function exhibits the asymptotic behavior G(r) ≈ (1/ r)exp(-r/r*) in the three-dimensional multichain model composed of Gaussian subchains without fixation of the mean-square length with orientational-deformational interactions.38,39 Therefore, eq A3 holds for this model only for a relatively small distance r < r* between subchains. Such a result may be because the chain bending rigidity and local interchain interactions play a significant part at small distances in the multichain model of Gaussian subchains as in the given model and that transversal correlations rapidly decay according to the exponential law at large distances of r > r* as in the single chain.26 The given chain composed of N rigid segments of a length l has the following mean-square projections of the vector h on the z- and y-axes Æh2 z æðNÞ ¼ Æh2 y æðNÞ=CN ln NþDN
ðA4Þ
where C = 1/(2πb) and D = 1 - C. Thus, a relatively slow decay of correlation between the fluctuations of transversal projections of segments (eq 7) in the considered model leads to the different dependence on the number of chain segments N as given by eq A4 compared with the Gaussian dependence Æh2æ ≈ N for the end-to-end distance of a single chain.26 The mean-square projection of a chain part composed of a sufficiently large number of segments N on the x-axis is defined by the relationship Æh2 x æðNÞ P P N l2 N n ¼1 n0 ¼1 Æcos γn cos γn0 æ and is calculated in the strongorder approximation as Æh2 x æ=L2 ½1 -ð2=NÞ
N X
fδdn gx =L2 ½1 -Æθn 2 æ=ð2=3Þð1 -S0 ÞL2
n ¼1
ðA5Þ In eq A5, L = Nl is the contour length of a chain part, angular fluctuations are given as Æθn2æ = 2Æ{δdn}xæ and S0 is defined by eq 5. (37) Pavlov, G. M.; Maximov, A. V.; Kusheva, I. V. Abstracts of European Polymer Congress, Portoroz, Slovenia, 2007; Poljansek, I.; Zigon, M., Eds.; Slovenian Chem. Soc.: Ljubljana, 2007; p 270. (38) Gotlib, Yu.Ya.; Baranov, V. G.; Maksimov, A. V. Vysokomol. Soedin., Ser. A 1985, 27, 312. (39) Ziman, J. M. Models of Disorder; Cambridge Univ. Press: London, 1979.
9092 DOI: 10.1021/la9005168
Appendix B: The Dependence of bc on the Chain Bending Rigidity (Parameter a) For flexible-chain polymers when the energetic parameters in potential eq 4 are of the same order of magnitude (K1 = K2),16 the critical value of the interchain interaction parameter in three-dimensional polymer systems with isotropic interactions between chain segments ( β = 1) is bc0 = 3C = 1.518 (point A in Figure 4), where C = I(1) = 0.506 is the value of the Watson integral.40 For chains with a sufficiently high rigidity and relatively small interchain interactions, β . 1 and the integral 6 is characterized by the asymptotic behavior: I( β) = 0.643/β. In this case, the critical value of the interchain interaction parameter bc is defined by the relation bc =3:721=a
ðB1Þ
The high asymptotic accuracy of the rigid-chain approximation (eq B1) at relatively small values of a g 4 was confirmed in the present study by the numerical solution of eq 8 (cf. curves 1 and 2 in Figure 8). From eq B1, it follows that, the higher the chain rigidity (a), the smaller the value of the interchain interaction parameter bc is at which the transition to the plane-ordered state takes place (Table 2, column2). For flexible-chain polymers at large interchain interactions, β,1 and another asymptotic equation I( β) = (1/π) ln(32/β) holds for integral 6. In this case, the critical value (eq 8) of the interchain interaction parameter bc is determined from the solution of the transcendental equation ðπbc =3Þ -ln bc ¼ lnð32=aÞ
ðB2Þ
The results of the numerical solution of this equation are also presented in Figure 8 (curve 3). In this limiting case of freely jointed chains (a = 0), the critical value is bc = ¥; that is, there is no phase transition to the plane-ordered state (Figure 8). At a > a* = 96/(πe) = 11.24, eq B2 has no solution at all, and the critical parameter bc is determined from eq B1, which holds at β g 1, that is, for rigid chains. At a < a*, the solution for eq B2 appears in the form bc = (3/π) |W(-1/(96πa))|, where W(x) is the Lambert function40 having two branches W(1, x) and W(2, x). At the branching point (a = a*), the solution for eq B2 is bc* = 3/π = 0.96. At a < a*, eq B2 has two roots bc1 and bc2, wherein bc2 < bc1. The solution bc2 (on the W(2, x) branch) corresponds to the region of β = a/b g 1 in which the flexible-chain approximation (eq B2) is already inapplicable. Furthermore, bc2 monotonically increases with an increase in chain rigidity (parameter a) and, thus, is physically meaningless. The second solution bc1 (on the W(1, x) branch) for eq B2 cannot be considered as a point of loss of isotropic-phase stability for flexible-chain polymers unless the chain rigidity parameter is a < a** = (3/π) ln 32 = 3.31 when the flexible chain approximation ( β = a/b e 1) is valid. The value of bc1 monotonically decreases with an increase in a as in the case of other approximations (Figure 8).
Appendix C: Measurements of Spontaneous Surface Birefringence In the theory of the tilted polarized-beam technique based on birefringence measurement,20-25 the surface birefringence B is (40) Jeffrey, D. J.; Knuth, D. E. Adv. Comput. Math. 1996, 5, 329.
Langmuir 2009, 25(16), 9085–9093
Maksimov et al.
Article
defined by the relationship )
B ¼ -ðπNA F=ðn3 λÞÞ½ðn2 þ2Þ=32 ½ða -a^ Þ=M0 S0 H0 ð1 -e -H=H0 Þ
ðC1Þ
)
where NA is Avogadro’s number, n is the refractive index of the polymer, λ is the wavelength of incident light, (a - a^) is the difference of the principal components of optical polarizability of the monomer unit, M0 is its molecular mass, F is the density of the polymer, H is the film thickness, and H0 is the total effective thickness of optically anisotropic film surface layers characterizing the distance from the film surface at which the orientational order parameter decreases by a factor of e relative to S0 (table 2, column 4).22,23 The measurements of spontaneous birefringence by means of the tilted incidence polarized-beam technique20,22 showed that
Langmuir 2009, 25(16), 9085–9093
the surface birefringence B for polysaccharides first linearly increases with an increase in the film thickness H and then tends to the limiting value B0 (Figure 6). Therefore, in accordance with eq C1, the experimental plots B(H ) in refs 22-24 were extrapolated by the relation BðHÞ ¼ B0 ½1 -expð -H=H0 Þ ¼ -cS0 ½1 -expð -H=H0 Þ
ðC2Þ
Here, c = B0/(-S0) is the fitting parameter, which was determined from experimental data (Table 1, columns 4, 6). Unlike polysaccharide films, for phenyl-containing sulfonated films the factor B ≈ -S, and birefringence does not show the tendency to saturation at increase in thickness H of film (Figure 7).
DOI: 10.1021/la9005168
9093
