ISSN 10637834, Physics of the Solid State, 2009, Vol. 51, No. 7, pp. 1365–1369. © Pleiades Publishing, Ltd., 2009. Original Russian Text © A.V. Maksimov, R.A. Gerasimov, 2009, published in Fizika Tverdogo Tela, 2009, Vol. 51, No. 7, pp. 1292–1296.
PROCEEDINGS OF THE XVIII ALLRUSSIA CONFERENCE ON PHYSICS OF FERROELECTRICS (VKSXVIII) (St. Petersburg, Russia, June 9–14, 2008)
Anisotropic Models of Polymer Ferroelectrics A. V. Maksimov* and R. A. Gerasimov Cherepovets State University, pr. Lunacharskogo 5, Cherepovets, Vologda region, 162600 Russia * email:
[email protected] Abstract—The phase transitions to an ordered state in threedimensional polymer systems that consist of flexible and rigid segments with intrachain and interchain orientational–deformational dipole interactions are considered. The statistic properties of the proposed models correspond to the Gaussian and spherical approximations that are used to describe the behavior of anisotropic Heisenberg ferroelectrics and ferromag nets. In the threedimensional models under consideration, there exists a critical point Tc where a second order phase transition from an isotropic state to a state with longrange orientational order occurs. The laws of variation in the temperature Tc as a function of anisotropy of the interactions are established by analytical methods. The temperature dependences of the longrange dipole order parameter for a given chain bending are calculated and the results are compared with the experimental data obtained by the piezoelectric pressure step (PPS) method for thick ferroelectric PVDF and P(VDF–TrFE) polymer films. PACS numbers: 61.30.Vx, 64.60.Ej, 64.70.M, 77.84.Nh DOI: 10.1134/S1063783409070117
1. INTRODUCTION In recent years, significant interest has been expressed in ferroelectric and ferroelastic polymer phases that exhibit unique electrical and mechanical properties [1–3]. These properties are associated with the onset of spontaneous polarization or deformation upon the phase transition from an isotropic state to an ordered state. Ferroelectric and ferroelastic materials are very promising for use in the design of radically new controlled signal processing devices (acousto electronic, optical, and electromechanical). The flexibility of polymer chains in combination with the ability of some polymers to undergo liquid crystal ordering has offered even wider opportunities for the manifestation and enhancement of the above properties [1–3]. Therefore, theoretical investigation of equilibrium and dynamic characteristics of polymer systems with ferroelectric and ferroelastic properties is an important problem. For this purpose, it is necessary to develop special models in which, on the one hand, dipole interactions between chain segments should be taken into account, as in standard models of ferroelec trics [4], and, on the other hand, chain segments should have the ability to undergo finite deformations (extension), like those, for example, in polymer net works [5–8]. Similarities revealed in ferromagnetic, ferroelec tric, and ferroelastic phase transitions with the use of experimental methods make it possible to develop common approaches to their analysis (for example, the mean field method), including polymer networks [1–3]. Statistics, dynamics, and specific features of phase transitions in nematic polymer systems can be studied using continuum and lattice models that con
sist of flexible or rigid segments, where a mean molec ular field of the quadrupole type is introduced. How ever, in order to describe orientation effects in polymer systems with ferroelectric and ferroelastic properties, we need models taking into account both the rigidity of chains and dipole interactions between chains. The class of such systems and conditions providing for their existence are discussed in [9]. 2. MODELS: SPHERICAL AND GAUSSIAN APPROXIMATIONS In the proposed threedimensional model, chains consist of rigid elements with a length l smaller than a statistic segment. It is assumed that N1 elements located at sites of a threedimensional “quasilattice” [10] along its longitudinal curvilinear direction (direc tion 1 in Fig. 1) form polymer chains. In Fig. 1, N2 and N3 characterizes the numbers of chains along each of the directions in the transverse cross section of the sys tem of chains. Therefore, the position of each segment in the quasilattice is determined by a set of three numbers (n1, n2, n3) ≡ n. The index n1 is counted along the outline of a given chain with the length N1l: n1 = 1, …, N1, and the indices n2 and n3 enumerate the chains themselves (along curvilinear directions 2 and 3 in Fig. 1): n2 = 1, … , N2; n3 = 1, … , N3. In this model, we introduce the poten tial of orientational interactions of the dipole type (i)
V n, m , which depends on the angle Φn, m of the mutual
1365
1366
MAKSIMOV, GERASIMOV
tem of chains that consist of rigid segments (Fig. 1) can be represented as
N1
V eff { u n, v n, w n } = –J 1
∑ ∑ ∑
( un um + vn vm + wn wm ) (2)
n1 – m1 = 1 n2 = m2 n3 = m3
n1
K2
– J2
K1
N3 n3 1
3 n2
∑
( u n u m + v n v m + w n w m ),
n1 = m1 n2 – m2 = 1 n3 – m3 = 1
K2
2
∑ ∑
N2
Fig. 1. Model of a threedimensional polymer system. The energy constant K1 describes the intrachain orientational interactions (along direction 1 of the quasilattice n ≡ (n1, n2, n3)). The constant K2 characterizes the interchain ori entational interactions (along directions 2 and 3 of the quasilattice).
spatial orientation between the axes of the segments located at n and m the sites of the “quasilattice”: ( un um + vn vm + wn wm ) (i) V n, m = – K i cos Φ n, m = – K i , (1) 2 l where un, vn, and wn are the orthogonal projections of the vector directed along the axis of the chain segment located at the n site of the threedimensional quasi lattice onto the coordinate axes (Fig. 1). In expression (1), the index i = 1 is used for segments belonging to the same chain and the index i = 2 is used for segments of the adjacent chains located in directions 2 and 3 in Fig. 1. The energy constant K1 along the longitudinal curvilinear direction 1 of the quasilattice describes the intrachain orientational interactions and is related to the thermodynamic rigidity of the chain [11], which determines the persistent length and the mean cosine of the angle between the adjacent elements of an isolated chain (for the case when K2 = 0). The corresponding constant K2 characterizes the local interchain orienta tional interactions, more specifically, the interactions between segments of different chains located in trans verse directions 2 and 3 of the quasilattice. The con stants K1 and K2 for polar and nonpolar macromole cules were estimated in [11]. Taking into account the interactions only between the nearest segments, the potential energy of the sys
where Ji = Ki /l 2 (i = 1, 2). According to the statistical properties, the proposed model with the effective potential energy of the dipole interaction between chain segments (expression (2)) should correspond to the threedimensional Heisenberg model with classi 2 2 2 cal spins of the fixed length ( u n + v n + w n = l 2). Therefore, even for the twodimensional isotropic variant of this model with potential (2), exact analyti cal calculations are impossible [11, 12]. Since the rotation group in the threedimensional space is non Abelian, the Gibbs distribution corresponding to potential (1), in contrast to the twodimensional model [11], cannot be reduced to the Gaussian form. However, as follows from various standard approxima tion methods (for example, from hightemperature expansions [12]), in the threedimensional isotropic Heisenberg model (K1 = K2 = K), there exists a critical temperature Tc ~ K/kB below which the system of magnetic dipoles (spins) is ordered. Therefore, it is quite reasonable to assume that, in the threedimen sional multichain system under consideration, the longrange orientational order can also exist below Tc. The use of simpler solvable variants of threedimen sional models with a harmonic interaction potential for a system of chains near the fully ordered orienta tion state [13] or a system of planar chains [14] con firms the above assumption. In this paper, we use the circumstance that the sta tistical properties of Heisenberg ferromagnets in the classical approximation are described fairly well by the threecomponent spherical model both at high and low temperatures [12, 15] (Fig. 2). Within the pro posed threedimensional model of a polymer system consisting of N1N2N3 rigid elements, as in the standard Berlin–Kac spherical approximation for the single component Ising model [12], it is assumed that not the square of the length of an individual rigid element 2 2 2 ( u n + v n + w n = l 2) is strictly fixed, but the quantity 1 N1 N2 N3
∑u
2 n
2
2
2
+ vn + wn = l .
(3)
n
The expression for the temperature Tc of the tran sition to an ordered state for threedimensional extended polymer systems can be reduced by the
PHYSICS OF THE SOLID STATE
Vol. 51
No. 7
2009
ANISOTROPIC MODELS OF POLYMER FERROELECTRICS
1367
methods employed in the spherical approximation [12,15] to the following form: 2
2K 1 2J 1 l T c = = , 3k B A ( ε ) 3k B A ( ε )
3
(4)
1
kBTc K1
2 T*c =
where the parameter ε = J2/J1 = K2/K1 characterizes the anisotropy of intrachain and interchain interac tions and
2 3 1
πππ
dψ 1 dψ 2 dψ 3 1 A ( ε ) = 3 . (5) 1 + 2ε – cos ψ 1 – ε cos ψ 2 – ε cos ψ 3 π 000
∫∫∫
0
In the absence of interchain interactions (ε = 0), integral (5) diverges. Hence, we have the critical tem perature Tc = 0 and, therefore, no phase transition occurs. In another multichain model, namely, the model of Gaussian subchains [16], the potential energy of the local intrachain and interchain orientational–defor mational interactions is describes by the quasielastic dipole potential V eff { u n, v n, w n } = 1 E 2 – E1
∑ ∑ ∑
∑ (u
2 n
2
0.2
0.4
ε
0.6
0.8
1.0
Fig. 2. Dependences of the reduced critical temperature T c* = kBTc /K1 on the ratio of the constants of intrachain and interchain interactions ε = K2/K1 = E2/E1: (1) within the mean field theory, (2) in the Gaussian multichain model of flexible segments with a fixed meansquare length for rigid chains α1 = E1/E0 Ⰷ 1 (ε Ⰶ 1), and (3) in the threecomponent spherical model of rigid chains with local intrachain and interchain interactions of the dipole type.
2
+ vn + wn )
n
( un um + vn vm + wn wm )
(6)
interactions E2). In particular, in the isotropic state, the following relationship must be satisfied:
n1 – m1 = 1 n2 = m2 n3 = m3 2
– E2
∑ ∑
∑
( u n u m + v n v m + w n w m ).
n1 = m1 n2 – m2 = 1 n3 – m3 = 1
The first term in expression (6) with the E constant is the energy of the quasielastic interaction intro duced within the Kargin–Slonimskiі–Rouse models [17]. This term describes the intrachain interactions due to the kinematic coupling between flexible ele ments of the chain. The second term in potential (6) with the E1 con stant, as in the Herst–Harris model [17], takes into account the intrachain rigidity, which formally corre sponds to the dipole–dipole interaction between the adjacent elements. The parameters E and E1 in poten tial (6) are related to the equilibrium characteristics of an individual chain. The third term in expression (6) with the E2 con stant describes the energy of local interchain orienta tional–deformational interactions of the dipole type. These constants were also estimated in [9]. Let us consider a uniform model where the condi tion of a fixed meansquare length l of an segment is introduced as follows: 2
2
2
2
〈 un 〉 + 〈 vn 〉 + 〈 wn 〉 = l .
No. 7
2
(8)
2
2E 1 l T c = , Dk B A ( ε )
(9)
where ε = E2/E1 = K2/K1 is the parameter of anisotropy of intrachain and interchain interactions and D = 2, 3 is the number of components of the vector describing an orientation of the chain segment. The expression obtained in [18], that is,
(7)
This conditions must be satisfied independently of the orientations of the chains (the constant of interchain Vol. 51
2
Within the threedimensional model of a polymer system consisting of flexible segments with a fixed meansquare length of the segments [16], the condi tion of the fixed meansquare length in the symmetric form of expression (8) for the isotropic state does not hold at the critical point Tc and in the subcritical region (T < Tc). This means that the threedimen sional system transforms into an ordered state with a preferred direction of orientation for chain segments, i.e., the ordering axis (which is called the “director” in liquidcrystal systems). The temperature Tc of the transition to the ordered state is determined by the expression [16]
2
PHYSICS OF THE SOLID STATE
2
〈 u n 〉 = 〈 v n 〉 = 〈 w n 〉 = l /3.
E i l = DK i , which relates the parameters of multi chain models consisting of rigid and flexible segments
2009
1368
MAKSIMOV, GERASIMOV
1.0 0.8 1 0.6 µ
2 0.4
3
0.2 0 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 T/Tc Fig. 3. Dependences of the longrange orientational order parameter µ on the ratio of the temperatures T/Tc for (1) the threedimensional isotropic spherical model (at ε = 1) and the Gaussian multichain model of flexible segments with a fixed meansquare length [16]. Points are experi mental data [19] obtained from the study of thermodepo larization by the piezoelectric pressure step (PPS) method in polymer films of (2) pure PVDF (polyvinylidene fluo ride) and (3) its copolymer P(VDF–TrFE) (vinylidene flu oride with tetrafluoroethylene). The corresponding dashed lines represent the results of the calculations within the charge trapping model (CTM) [19].
(i = 1, 2) for ε Ⰶ 1, makes it possible to represent expres sion (9) in the form Tc ≅ 3.12 K 1 K 2 /kB
(10)
(because A(ε) ≅ 0.643/ ε ), where K1 and K2 are con stants of local intrachain and interchain interactions of rigid segments. In the limiting case, when inter chain interactions are absent (E2 = 0), we have the critical temperature Tc = 0. Therefore, no phase tran sition to an ordered state in an isolated chain occurs. 3. THEORY AND EXPERIMENT Figure 3 shows the dependences of the longrange orientational order parameter µ on the ratio of the temperatures T/Tc for the above models of polymers. It can be seen that, in all cases, as temperature increases, the parameter µ gradually monotonically decreases to zero, which is reached at the critical point, i.e., at the Curie temperature Tc. That suggests that, in these systems, there occurs a secondorder phase transition from an ordered state to an isotropic state. The results of the analytical calculations of the longrange dipole order parameter for the model of flexible segments [16] is in qualitative agreement with experimental data and theoretical calculations [19] of the temperature dependence of the remanent polar
ization in polymer electret films (Fig. 3). In these experiments, it was taken into account that the revers ible component of the polarization decreases because of the pyroelectric effect. It is worth noting that the contribution of this component was measured by com paring the remanent polarization revealed during heating from 30 to 180°C with its value after polariza tion at room temperature and after repeated cooling to room temperature. The experiments performed in [19] demonstrated that, up to 150°C, the decrease in the reversible polarization is small compared to the irre versible component. A significant effect was found when unpolarized PVDF samples were heated to 180°C (i.e., to the melting point of the crystals). In this case, an anomalous increase in the polarization was observed during cooling to room temperature. This effect is explained by a partial melting and recrystalli zation of crystallites [19]. Figure 3 also presents the results of the experimental study (the piezoelectric pressure step (PPS) method) of the thermodepolariza tion of pure PVDF (curve 2) and its copolymer P(VDF–TrFE) of the 65/63 composition (curve 3). The samples were prepared in polarizing fields of 1.6 × 108 and 0.6 × 108 V/m for 1 and 2 min, respectively. The polarization was performed at room temperature. As can be seen from Fig. 3, the polarization of PVDF decreases over a wide temperature range (~150°C) from room temperature to the melting point of crystal lites. In contrast to PVDF, the remanent polarization of the copolymer decreases in a substantially narrower temperature range. 4. CONCLUSIONS Experimental values of the remanent polarization at sufficiently low (room) temperatures turned out to be somewhat higher than the values calculated within the proposed model. This can be explained by the effect of nonequilibrium conditions of the experiment, which are ignored within the model [16], in contrast to the theoretical approach used in [19], where these effects are considered and, therefore, agreement between the experimental data and theoretical calcu lations is much better. REFERENCES 1. A. Merenga, S. V. Shilov, F. Kremer, G. Mao, Ch. K. Ober, and M. Brehmer, J. Phys. II 4, 859 (1994). 2. E. M. Terentjev and M. Warner, J. Phys. II 4, 849 (1994). 3. S. V. Sholov, H. Skupin, F. Kremer, E. Gebhard, and R. Zentel, Liq. Cryst. 22, 203 (1997). 4. V. G. Vaks, Introduction to the Microscopic Theory of Ferroelectrics (Nauka, Moscow, 1973) [in Russian]. 5. M. Brehmer and R. Zentel, Liq. Cryst. 21, 589 (1996). 6. P. I. Teixeira and M. Warner, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 60, 603 (1999).
PHYSICS OF THE SOLID STATE
Vol. 51
No. 7
2009
ANISOTROPIC MODELS OF POLYMER FERROELECTRICS 7. E. M. Terentjev and M. Warner, J. Phys.: Condens. Matter 11, 239 (1999). 8. Yu. Ya. Gotlib and A. A. Gurtovenko, Macromolecules 33, 6578 (2000). 9. A. V. Maksimov, O. G. Maksimova, and D. S. Fedorov, Vysokomol. Soedin., Ser. A 48 (7), 1151 (2006) [Polym. Sci., Ser. A 48 (7), 751 (2006)]. 10. A. V. Maksimov, Vysokomol. Soedin., Ser. A 50 (3), 518 (2008) [Polym. Sci., Ser. A 50 (3), 341 (2008)]. 11. A. V. Maksimov and O. G. Maksimova, Vysokomol. Soedin., Ser. A 45 (9), 1476 (2003) [Polym. Sci., Ser. A 45 (9), 859 (2003)]. 12. R. Baxter, Exactly Solved Models in Statistical Mechan ics (Academic, London, 1982; Mir, Moscow, 1985). 13. Yu. Ya. Gotlib, V. G. Baranov, and A. V. Maksimov, Vysokomol. Soedin., Ser. A 29, 2620 (1987).
PHYSICS OF THE SOLID STATE
Vol. 51
No. 7
1369
14. A. V. Maksimov and Yu. Ya. Gotlib, Vysokomol. Soe din., Ser. A 30, 1411 (1988). 15. É. L. Nagaev, Fiz. Tverd. Tela (Leningrad) 27 (11), 3270 (1985) [Sov. Phys. Solid State 27 (11), 1970 (1985)]. 16. A. V. Maksimov, Vysokomol. Soedin., Ser. A 49 (5), 891 (2007) [Polym. Sci., Ser. A 49 (5), 599 (2007)]. 17. Yu. Ya. Gotlib, A. A. Darinskiі, and Yu. S. Svetlov, Physical Kinetics of Macromolecules (Khimiya, Lenin grad, 1986) [in Russian]. 18. Yu. Ya. Gotlib and A. V. Maksimov, Vysokomol. Soe din., Ser. A 34, 157 (1992). 19. A. A. Rychkov, D. A. Rychkov, and S. A. Trifonov, Dielectric Polymers (Knizhnyі Dom, St. Petersburg, 2005), p. 118 [in Russian].
2009
Translated G. Tsydynzhapov
