Phase Diagram of Diblock Copolymer Melt in Dimension d = 5

Transcription

1 COMPUTTIONL METHODS IN SCIENCE ND TECHNOLOGY 7(-2), 7-23 (2) Phase Diagram o Diblock Copolymer Melt in Dimension d = Faculty o Physics,. Mickiewicz University ul. Umultowska 8, - Poznań, Poland mbanasz@amu.edu.pl (eceived: 2 November 2; revised: December 2; accepted: 9 December 2; published on-line: December 2) bstract: Using the sel-consistent ield theory (SCFT) in spherical unit cells o various dimensionalities, D, a phase diagram o a diblock, -b-, is calculated in dimensional space, d =. This is an extension o a previuos work or d =. The phase diagram is parameterized by the chain composition,, and incompatibility between and, quantiied by the product χn. We predict stable nanophases: layers, cylinders, 3D spherical cells, D spherical cells, and D spherical cells. In the strong segregation limit, that is or large χn, the order-order transition compositions are determined by the strong segregation theory (SST) in its simplest orm. While the predictions o the SST theory are close to the corresponding SCFT extrapolations or d =, the extrapolations or d = signiicantly dier rom them. We ind that the S nanophase is stable in a narrow strip between the ordered S nanophase and the disordered phase. The calculated orderdisorder transition lines depend weakly on d, as expected Key words: diblock copolymer melt, phase diagram, order-disorder transition, order-order transition, ield theory I. INTODUCTION Diblock copolymer (DC), -b-, melts consist o 2 types o segments, and, arranged in 2 corresponding blocks. Those melts can sel-assemble in 3d into various spatially-ordered nanophases, such as layers, (L), hexagonally packed cylinders, (C), gyroid nanostructures, (G), with the Ia3d symmetry, and cubically packed (either bodycentered or closely packed) spherical cells (S), depending on the chain composition, ( is the raction o -segments; is the raction o -segments), degree o polymerization (number o segments), N, and the temperature-related χ parameter [, 2]. ecently, an additional O 7 -phase has been reported [3, ], but it is stable in a very small region o the phase diagram. Those nanophases can be transormed into a disordered phase, or example, upon heating. It is o great interest to determine a phase diagram o such melts exhibiting order-disorder transition (ODT) lines, also reerred to as binodals o microphase separation transition (MST), and order-order transition (OOT) lines. This task has been largely achieved or 3-dimensional (bulk) diblock melts by accumulating results rom numerous experimental and theoretical studies [-2], also or 2d diblock copolymer melts [3]. The L, C, and S nanophases are known as classical, whereas G and O 7 nanophases are reerred to as nonclassical, or sometimes complex. The Wigner-Seitz cell o a classical phase can be approximated by D-dimensional sphere, S D, both in the real r -space and the reciprocal k -space. Within this approximation, known as Unit Cell pproximation (UC), the L, C, S nanophases correspond to S, S 2, and S 3, respectively, and the spacial distribution o chain segments can be mapped with a single radial variable, r, as shown in Table. The classical phases can be

2 8 Table. Unit cell equations o D-dimensionality; equations are supplemented with the unconstrained variables or corresponding d s (2, 3, and ) L (S ) Nanophase Cell equation d = 2 d = 3 d = d = adial coordinate x < y y, z y, z, t y, z, t, v r = x C (S 2 ) 3 S 3 S S x + y < * z z, t z, t, v x + y + z < imp * t t, v x + y + z + t < imp imp * v x + y + z + t + v < imp imp imp * 2 2 r = x + y r = x + y + z r = x + y + z + t r = x + y + z + t + v *indicates the absence o unconstrained variables; imp indicates that the nanophase or this d is impossible easily generalized to higher dimensions, in particular or d = we can have nanophases S D, with dimensionality, D, ranging rom to. It is interesting that a mean-ield (MF) theory applied to copolymer melts [8,, ], known as the Sel-Consistent Field Theory (SCFT), is successul in predicting diblock phase diagrams resembling the experimental ones, as shown, or example, in e. []. The SCFT approach is based on the assumption that coarse-grained polymer chains in dense melts are Gaussian (Flory's theorem [7]), and on the MF approximation which selects the dominant contribution in the appropriate partition unction, thus neglecting luctuations. ecause, in the MF theories, it is suicient to know the composition,, and the product χn in order to oresee the nanophase [,, 8], the diblock phase diagram can be mapped in (, χn)-plane. The MF theories exist in many variations, both in real space ( r -space) [, 7, 9] and Fourier space ( k -space) [2, 8]. In addition, we intend to compare the phase boundaries calculated by the SCFT (and extrapolated to the strong segregation limit) with the strong segregation theory (SST) or diblock melts, developed by Semenov [9], in which the ree energy o the nanophase has three contributions, the interacial tension and the stretching (o entropic origin) energies o the and blocks. These energies can be approximated by simple expressions, allowing the calculation o the OOT compositions in the SST. The main goal o this paper is to construct a phase diagram o a copolymer melt or d =, applying the SCFT method with the UC in r-space, as presented in [-8]. Speciically, we intend to determine the area in (, χn)-space, in which the S phase is stable, by varying both the radius,, o the unit cell and the dimensionality, D. In previous work [2] we calculated the phase diagram o the diblock copolymer melt or d =, and managed to answer the ollowing questions:. is the S nanophase stable? 2. what is the sequence o nanophases upon changing? 3. are the binodals (ODT lines) shited as we vary d rom to?. what are the strong segregation limits o the OOT lines or d =? The answers were as ollows:. the nanophase S is stable within a relatively narrow strip between the S 3 nanophase and the disordered phase, 2. the sequence o nanophases appropriate or the UC in 3d is preserved, starting rom = /2, L, C, S 3, and there is an additional S nanophase in d, 3. the ODT binodals depend weakly on d, and they are shited as d is varied,. the SST compositions, LC /, CS /, and S are close to the corresponding extrapolations rom the sel- consistent ield theory. In this work, as in e. 2, the ollowing questions are posed:. is the S nanophase stable? 2. what is the position o the S nanophase in sequence o phases upon changing? 3. are the binodals (ODT lines) shited as we vary d rom to?. what are the strong segregation limits o the OOT lines or d =? II. METHOD The incompressible copolymer melt is modeled as a collection o n diblock chains conined in volume V. Each

3 Phase Diagram o Diblock Copolymer Melt in Dimension d = 9 chain, labeled =, 2,..., n, can take any Gaussian coniguration (in accordance with the Flory's Theorem [7]) parameterized rom s = to s = or -segments, and rom s = to s = or -segments. Up to a multiplicative constant, the partition unction or a single Gaussian chain in external ields W (r) and W (r) acting on segments and, respectively, is exp [ W, W ]/ D r () ( r ( )) r ( ) ( ) dsw s dsw s The path integral, D r (), is taken over single-chain trajectories, r (s), with Wiener measure expressed as D r [ ] = D r P r ;,, and s2 2 3 d P[ r; s, s2] exp ds ( s) 2 2Na r (2) ds s Note that a is the segment size, and Na 2 is the mean squared end-to-end distance o a Gaussian chain. y Kac- Feynman theorem, Eq. can be related to a Fokker-Planck partial dierential equation [2], known also as a modiied diusion equation (MDE) and shown with appropriate details below (Eqs. and ). Segments and interact via the χ parameter which provides an eective measure o incompatibility between them [7]. Evaluation o the ull partition unction o n interacting diblock chains, shown below (Eq. 3), is a highly challenging task, involving many-body interactions, both intermolecular and intramolecular. n Z = D r [ ˆ ˆ ] exp ˆ ˆ δ φ φ χρφφ (3) = where δ-unction enorces incompressibility (the melt is assumed to be incompressible), and n ρ = ( s ) () ˆ N φ () r = ds δ r r () () decomposing the appropriate δ-unctionals, the partition unction o an incompressible diblock melt is () () φ () () () Z = N Dφ DW D DW D Ψ [ φ,, φ,, Ψ] F W W exp kt where N is a normalization actor. The unctional integral is taken over the relevant ields φ(), r W (), r φ(), r W (), r and Ψ( r ), with the ree energy unctional, F[ φ, W, φ, W, Ψ], including the single chain partition unction (in external ields W (r) and W (r)), as shown below [ φ,, φ,, ] F W W Ψ + nk T () ln V d Nχφ( ) φ( ) + V r r r ( ) φ ( ) W ( ) φ ( ) W r r r r + (7) ( r) ( ( r) ( r )) φ φ Ψ Fields φ ( r ) and φ ( r ) are associated with normalized concentration proiles o and, and ields W (r) and W (r) with chemical potential ields acting on and, respectively; ield Ψ(r) enorces incompressibility. Evaluating unctional integrals in Eq. is a challenging task which, in principle, can be perormed by ield theoretic simulations as proposed and implemented by Fredrickson and coworkers [2, ]. simpler but approximate approach is based on the mean-ield idea, where the dominant, and in act only, contribution to the unctional integral in Eq. comes rom the ields satisying the saddle point condition expressed as the ollowing set o equations: δf δf δf δf δf = = = = = δφ δφ δw δw δ Ψ Perorming the above unctional derivatives yields (8) W () r = Nχφ () r +Ψ() r (9) W ( r) = Nχφ ( r) +Ψ( r ) () = φ ( r) + φ ( r ) () n ˆ N φ () r = ds δ r r () () ρ = ( s ) V φ () r = ds q(,) r s q (,) r s (2) are the microscopic segments densities o and, respectively; ρ = nn/v is the segment number density. ter replacing microscopic segment (or particle) densities with a variety o ields [2, -8], by inserting and spectrally φ = where /V can be calculated as V () ds q (,) s q r r (,) r s (3)

4 2 = d q(,) V V r r and q(r, s) is the orward chain propagator which is the solution o the ollowing modiied diusion equation q Na q W q s = 2 2 ( r), q = 2 2 Na q W r q s ( ), () with the initial condition q(r, ) =. Similarly q (,) r s is the backward chain propagator which is the solution o the conjugate modiied diusion equation: q 2 2 = Na q W() r q, s () q 2 2 = Na q W () r q, s with the initial condition q (,) r =. While the set o Eqs. 9,,, 2, and 3 can be solved, in principle, in a sel-consistent manner, it is diicult to solve this set without some additional assumptions. Firstly, we assume that the melt orms a spatially ordered nanophase. Secondly, we use the UC which is a considerable simpliication, limiting our attention to a single D-dimensional spherical cell o radius, and volume V. ll ields, within this cell, have radial symmetry, which reduces this problem computationally to a single radial coordinate, r. The unconstrained spatial variables, speciied in Table or each d, become computationally irrelevant. Thus Eq. can be rewritten as = D V D r q(,) r dr D (7) Note that the actor, D, in ront o the above integral originates rom the ratio o the area o a sphere with radius to the volume o a spherical cell with the same radius, both in D dimensions. While in integrals (Eqs. 2, 3 and ) we replace r with r, and dr/v with Dr D dr/ D, in the modiied diusion Equations, and, we replace r with r and use the spherically symmetric orm o the Laplacian 2 2 D = + 2 r r r (8) and similarly, in equations or both propagators q(r, s) and qrs (,), we replace r with r. Obviously the solution depends on radius,, and dimensionality, D =, 2, 3, and, corresponding to dierent nanophases, shown in Table. We use the Crank-Nicholson scheme [2] to solve iteratively the modiied diusion equations (Eqs. and ) in their radial orm, until the sel-consistency condition is met, obtaining the saddle point ields, φ( r), φ ( r), W( r) and W ( r ) or a given and D. In the MF approximation, the ree energy unctional becomes the ree energy, and thereore we calculate the reduced ree energy (per chain in k T units) by substituting the saddle point ields into Eq. 7: F(, D) ln + nk T V (9) D D () () () () () ( ) D r N χφ r φ r W r φ r W r φ r dr + III. ESULTS ND DISCUSSION Since in the MF theory, the stability o a nanophase depends on the product χn and composition,, we start at a given point o the phase diagram (, χn) with a numerical calculation o F(, D) (Eq. 9) or various D s (, 2, 3,, and ) and s. In order to solve the MDE's (Eqs. and ) we use up to N T = and up to N = 8 steps or the time, s, and space, r, variables, respectively. Numerically, we ind and D which minimize F(, D), and this allows us to determine the dimensionality, D, o the most stable nanophase, and thereore the most avorable nanophase itsel, using the correspondence rom Table. However, the ree energy o this nanophase has to be compared to that o the disordered phase. Thereore, we calculate the dierence ΔF F Fdis / (2) nkt nkt nkt where F dis is the ree energy o the disordered phase: F = N ( ) (2) nk T dis χ I ΔF is negative then the appropriate nanophase is thermodynamically stable or the point considered, (, χn); otherwise the system is the disordered phase. For example, in Figure we compare ΔF/(nk T) or d = and d = as a unction o, at χn =. The intersection o those ree energy curves occurs at S/ S =.8, as also indicated in Table 2. This procedure allows us to map the DC melt phase diagram or d = in the (, χn)-plane, as shown in Fig. 2. Since there is a mirror symmetry with respect to =.

5 Phase Diagram o Diblock Copolymer Melt in Dimension d = 2 (, can be exchanged with ), we show the resultant nanophases only rom = to., and the ollowing phase sequence is observed: L, C, S 3, S, S and the disordered phase; the corresponding data or those lines is presented in Table 2. new nanophase, S, is observed in a relatively narrow strip between the S phase and disordered phase. parameter. The resultant limits, LC /, CS /, S, determined in e. 2 and S/ S (determined in this paper) are compared to the SST s, as shown in Table 3. The discrepancy between the SST and the present SCFT with the UC, or LC / and CS / is within 2% error, as also 3 reported in [9], and the discrepancy or S3/ S is about %. However, the dierence between the extrapolated S/ S =. and the calculated S/ S =.8 (rom SST) is much larger. Since the SCFT is more advanced and accurate theory than the SST (the SCFT is a ull mean ield theory, and the SST is an approximation o the mean ield theory which is meaningul only at strong segregations), we demonstrate that the area o stability or the S phase is mostly likely to be signiicantly larger than that predicted rom the SST. Fig.. χf (in nk T units) as a unction o or χn =. ed line indicates the results or the S nanostructure and green line or the S nanostructure. The lines intersect at S /S =.8 Table 2. The ODT and OOT lines or selected χn s N χ LC / CS / 3 S S 3/ S S ODT / This is the main result o this paper. We extrapolate the calculated OOT lines, LC /, CS /, S, S/ S to the strong segregation limit, that is, we estimate them as χn (or /(χn) ), itting to the ollowing unction: ( χ ) g N = + (22) χ N as used in reerenced [9] and [2], where is the extrapolation to the strong segregation limit, and g is a itting Fig. 2. DC phase diagram in d, L, C, S 3, S, and S indicate corresponding nanophases; the disordered phase is also shown Table 3. The OOT lines rom the ull SCFT [9] and UC extrapolated to ininite χn s comparedto the SST results Method LC / CS / 3 S / S 3 ull SCFT.3. S / S UC SST While spinodals or the ODT calculated with randomphase approximation (P) [] are the same or d = 2, 3, and, the binodals (the ODT lines), calculated in this work, depend on d as shown in Fig. 3. The binodals depend weakly on d, and they are particularly close to each other in the vicinity o = /2 (symmetric diblock), and thereore

6 22 the previous study [2], the S/ S extrapolation does not agree with the SST predictions. We ind that the S nanophase is stable in a narrow strip between ordered S nanophase and the disordered phase. The calculated binodals (ODT lines) depend weakly on d, as expected. cknowledgments We acknowledge a computational grant rom the Poznan Supercomputing and Networking Center (PCSS). eerences Fig. 3. The ODT lines or d = 2, 3, and. The inset is rom χn = 7 to 7 we show them in a narrow window (rom 7 to 7 in χn, that is away rom = /2) in the inset o Fig. 3. We observe the sequence o binodals, as shown in the inset o Fig. 3. For = /2 the P spinodal is at (χn) c.99, and the calculated binodals (or d = 2, 3, and ) also converge to this point within the numerical accuracy. Similarly, the OOT lines seem to converge to (χn) c or =.. IV. CONCLUSIONS Using the sel-consistent ield theory in spherical unit cells o various dimensionalities, D =, 2, 3, and, we calculate the phase diagram o a diblock, -b-, copolymer melt in -dimensional space, d =. The phase diagram is parameterized by the chain composition,, and incompatibility between and, quantiied by the product χn. We predict stable nanophases: layers, cylinders, 3D spherical cells, D spherical cells and D spherical cells, and calculate both order-disorder and order-order transition lines. In the strong segregation limit, that is or large χn, the OOT compositions, LC /, CS /, S and S/ S are determined by the strong segregation theory. While LC /, CS /,and S are close to the corresponding extrapolations rom the sel-consistent ield theory, as shown in [] I.W. Hamley, Developments in lock Copolymer Science and Technology (John Wiley & Sons, erlin, 2). [2] G.H. Fredrickson, The Equlibrium Theory o Inhomogeneous Polymers (Clarendon Press, Oxord, 2). [3] T.S. ailey, C.M. Hardy, T.H. Epps, F.S. ates, Macromolecules 3, 77 (22). [] M. Takenaka, T. Wakada, S. kasaka, S. Nishisuji, K. Saijo, H. Shimizu, M.I. Kim, H. Hasegawa, Macromolecules, 399 (27). [] L. Leibler, Macromolecules 3, 2 (98). [] M. anaszak, M.D. Whitmore, Macromolecules 2, 3 (992). [7] J.D. Vavasour, M.D. Whitmore, Macromolecules 2, 77 (992). [8] M.W. Matsen, M. Schick, Phys. ev. Lett. 72, 2 (99). [9] M.W. Matsen, M.D. Whitmore, J. Chem. Phys., 998 (99). [] M.W. Matsen, J. Chem. Phys., 28 (2). [] E.M. Lennon, K. Katsov, G.H. Fredrickson, Phys. ev. Lett., 3832 (28). [2] T. Taniguchi, Journal o the Physical Society o Japan 78, 9 (29). [3] I.W. Hamley, Progress in Polymer Science 3, (29). [] M.W. Matsen, in Sot Condensed Matter, Vol., edited by G. Gompper and M. Schick (John Wiley & Sons, erlin, 2). [] E.W. Cochran, C.J. Garcia-Cervera, G.H. Fredrickson, Macromolecules 39, 29 (2). [].K. Khandpur, S. Forster, F.S. ates, I.W. Hamley,.J. yan, W. ras, K. lmdal, K. Mortensen, Macromolecules 28, 879 (99). [7] P.G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 979). [8] M.W. Matsen, J. Phys.: Condens. Matter, 2 (22). [9].N. Semenov, Sov. Phys. JETP, 733 (98). [2] M. anaszak,. Koper, K. Lewandowski, P. Knychala, Submitted.

7 Phase Diagram o Diblock Copolymer Melt in Dimension d = 23 MICHŁ DZIĘCIELSKI, received his MSc degree in Physics in 2 rom the dam Mickiewicz University in Poznań where he is a PhD student in Physics. His research interests concern polymer physics, parallel algorithms, chemical reactions and programming. KZYSZTOF LEWNDOWSKI, received his MSc degree in Computer Science in 27 rom the dam Mickiewicz University in Poznań where he is a PhD student in Physics. His research interests concern polymer physics, globular and micellar nanostructures, parallel algorithms and programming. MICHŁ NSZK, graduated in 98 rom dam Mickiewicz University in Poznan, Poland. He received his PhD degree in Physics in 99 rom Memorial University in St. John's, Canada. From 992 to 99 he worked as postdoctoral ellow in Exxon esearch & Engineering Co. in nnandale, New Jersey, US, and rom 99 to 997 in the Chemistry Department o UMIST in Manchester, UK. In 997 he joined the dam Mickiewicz University, obtaining the DSc degree (habilitation) in 2 in Physics. His main interest is in developing new models and theories or nanoscale sel-assembly o various copolymer systems. COMPUTTIONL METHODS IN SCIENCE ND TECHNOLOGY 7(-2), 7-23 (2)