Simulation of the collective dynamics scattering of diblock copolymers with heterogeneous composition in the semidilute regime

Journal of Non-Crystalline Solids 352 (2006) 5060–5066 www.elsevier.com/locate/jnoncrysol

Simulation of the collective dynamics scattering of diblock copolymers with heterogeneous composition in the semidilute regime Esteban Rodrı´guez, Juan J. Freire

*

Departamento de Ciencias y Te´cnicas Fisicoquı´micas, Facultad de Ciencias, Universidad Nacional de Educacio´n a Distancia, 28040 Madrid, Spain Available online 22 August 2006

Abstract The dynamic behavior of symmetric diblock copolymer chains in semidilute solution with heterogeneous composition is simulated by means of the bond fluctuation model with repulsive interactions between monomers of different types. We have considered two different lengths at a given concentration close to the order–disorder transition. We have computed the radius of gyration, the translational diffusion coefficient and the first Rouse relaxation time, but most of the analysis is devoted to the results for the collective dynamic scattering functions. As in the simpler case of copolymers without compositional heterogeneity, we can fundamentally distinguish two modes in these functions, but the behavior of the main, slower, mode is clearly altered by heterogeneity. According to the present simulations, this mode reflects the slowest Rouse internal motion together with a purely diffusive heterogeneity contribution. In some cases, the two contributions of the main mode can be separated. However, the retardation of the slower mode in the proximity of the intensity maximum and other subtle features predicted by the random phase approximation theory are not clearly visible in the simulation data from these particular systems. In most cases, we have found a less intense and significantly faster mode, also appearing in previous simulations for which the heterogeneity feature was not included. The discussion also takes into account existing experimental data for diblock copolymer solutions. Ó 2006 Elsevier B.V. All rights reserved. PACS: 82.35.Jk; 83.80.Uv Keywords: Monte Carlo simulations; Polymers

1. Introduction The peculiar properties of diblock copolymer solutions have for some time been subject of much attention [1]. In particular, the order–disorder transition (ODT) has been extensively studied by applying the self-consistent field theory [2] or mean field theory with fluctuations [3], developed some time ago. Alternative more recent theories avoid using the random phase approximation (RPA) [4] and show a complex behavior from the fully disordered to the *

Corresponding author. Tel./fax: +34 913988627. E-mail address: [email protected] (J.J. Freire).

0022-3093/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.03.121

fully segregated state. These theories were initially proposed for melts. However, semidilute solutions may similarly approach to an order–disorder transition (ODT) for increasing values of concentration in a common good solvent and, in fact, the fluctuation effects in the mean field theory have been reformulated in the context of the blob theory for semidilute solutions [5]. Scattering techniques are particularly useful in exploring the structural and dynamic properties of these systems [6], since the different contrast factors (or refractive indices) of the blocks relative to the solvent yield scattering functions that differ substantially from those associated with homopolymer chains. Thus, the total scattering intensity shows a maximum at

E. Rodrı´guez, J.J. Freire / Journal of Non-Crystalline Solids 352 (2006) 5060–5066

an intermediate value q* of the scattering variable, q. This maximum tends to diverge when the systems are close to the ODT [3]. The dynamics scattering functions are typically multiexponential, showing the contribution of several modes. In a very basic description, a first mode should be associated with the main internal motion of the polymer [7], i.e. it is related to the relaxation time of the first Rouse mode, s1. Its decay rate or frequency can be expressed, at low q, as Cint = 1/s1. Assuming that the Rouse theory for single chains can also qualitatively describe this feature, it is expected that there is a contribution depending on the translational diffusion coefficient of individual chains, D, for moderate values of q, so that Cint = 1/s1 + Dq2 with this correction [8]. These conclusions can only be applied for non-segregating systems. Using the RPA theory it has been shown, however, that the internal mode suffers a retardation (associated with the increase of the scattering intensity) for systems approaching to the ODT [9]. Diblock copolymers always exhibit some degree of compositional heterogeneity as a result of the (usually small) polydispersity of the two homopolymer chains from which they are assembled [10]. It has been recognized that this amount of compositional heterogeneity is manifested by the presence of an additional translational-dependent heterogeneity mode (in its most basic description for low q) whose frequency is given by Chet = Dq2. A considerably more elaborated scheme can be obtained from the RPA, but in many cases it can be understood in terms of this simple description. When the system is sufficiently close to the ODT so that both the basic ‘internal’ and ‘heterogeneity’ contributions tend to intercept at intermediate q (due to the alteration of the internal contribution, manifested by the increase of intensity or the retardation of frequency) there is an ‘eigenvalue effect’ so that the curves corresponding to the ‘slow’ and ‘fast’ mode do not intersect each other [11]. The resulting pattern is shown in Fig. 9 of Ref. [11]. This scheme is in agreement with detailed dynamic light scattering experiments performed for diblock copolymer solutions. In addition to these features, other modes may appear for most systems. A collective mode, describing faster local fluctuations in the polymer–solvent composition around the entanglement points is present in all types of semidilute solutions of homopolymers [6] and should also appear for copolymer solutions except in the case that the refractive index of solvent has been exactly set to the so called ‘zero-average contrast conditions’ [12]. This mode is independent of the polymer length and its frequency shows a diffusive (q2-dependent) behavior. Also, there is some indication that faster internal motions (correlated to Rouse modes of higher orders) may separate giving a non-diffusive (q independent) contribution, usually detected as a single mode [13]. Although there is a basic agreement between theory and experimental data for the scattering of these systems, their complementary study by means of simulation methods is

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recommended for several reasons: (a) the simulations allow for a simpler set of the interesting values of the variables involved, being in fact the only type of ‘experiments’ available for some ideal systems (for instance, they allow for a direct comparison with perfectly symmetric diblock systems without compositional heterogeneity, which cannot be investigated experimentally). (b) It is necessary to know the performance of different simulation methods and models in relation with this complex property, since the correct reproduction of the systems whose behavior is known may increase the motivation to perform simulations for other systems that cannot be simply described by theory (such as more complex types of block polymers, solutions with a selective solvent, etc.). A great variety of numerical simulations including the use of conventional [14] and discontinuous molecular dynamics [15], dissipative particle dynamics [16], time-dependent Ginzburg–Landau equations [17], and several varieties of Monte Carlo methods [18,19] have been developed to study the phase behavior of diblock copolymer melts. The bond fluctuation model has been used to simulate a great variety of polymer systems, since it combines the numerical advantages of conventional lattice models, whilst closely mimicking the continuum behavior, being therefore appropriate to study the polymer dynamics when a simple ‘bead-jump’ algorithm is employed [20,21]. Therefore, though the model may be less accurate in the local description of the polymers and does not include hydrodynamic interactions (which are especially important in dilute systems) we consider it is particularly valuable with respect to capturing the essential physical behavior of more concentrated solutions. In previous studies we have usefully employed this model in the investigation of the dynamic scattering of solutions, particularly semidilute solutions of linear homopolymers and diblock copolymers without compositional heterogeneity [22]. Specifically for the latter systems, we have considered symmetric diblock copolymers, where each block consists of N/2 beads of either type A or type B with opposite contrast factors in a non-selective solvent (represented by the lattice voids). In order to induce block segregation, a model with net repulsion between interacting neighbors for beads of different types was considered. Pairs of beads of the same type, i.e. AA or BB, did not interact. With this model, we have obtained collective dynamic scattering curves exhibiting a slow mode with decay rates apparently close to the Cint = 1/s1 + Dq2 behavior. This mode shows an apparent though non-totally conclusive retardation at intermediate values of q, and a very clear increase of the intensity at the same range of q values. We also found a faster mode, apparently non-diffusive and with its intensity maximum at intermediate q, that seems to reflect the accumulation of faster Rouse motions. Furthermore, we have approximately characterized the ODT transition curves for the same model [23]. Some time ago, we also studied the individual form factors of similar diblock copolymers with a simple cube lattice without including segregation effects (purely

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2. Model and methods The simulation model and algorithm has been described in detail in our previous work [22]. We consider n selfavoiding walk (SAW) chains, each composed of N beads in a cubic lattice of length L, adequately chosen according to the predicted mean chain size, including periodic boundary conditions. We choose two different values of the chain lengths, N = 40 and 100 and assign the box size values of L = 76 and L = 100, respectively. The distance between adjacent sites, b, is taken as the length unit. Each one of these beads blocks the 26 surrounding lattice sites. These sites cannot be occupied by other beads, according to the SAW condition. Furthermore, other additional beads (depending on the site position) can simultaneously block some of the same 26 sites. Accordingly, it is easy to verify that a single bead effectively blocks a total of eight sites. Consequently, the required polymer volume fractions is U = 8nN/L3. Elementary bead jumps are achieved by randomly moving a bead to one of the six closest sites. This way we generate ‘dynamic’ Monte Carlo trajectories from previously equilibrated samples. In the trajectories, a time unit is constituted by nN subsequently attempted bead jumps. Once a move is attempted, the new bond lengths are checked and the SAW condition is enforced. In order to do so, we verify that the new bead position is not blocked by other beads. Otherwise, the new configuration is rejected. In addition to the SAW condition, we adopt the model proposed by Wittkop et al. to include interactions between non-neighboring beads [25]. In the present model, we just consider a repulsive interaction between A and B units of the copolymer systems, expressed in terms of the energetic parameter b  e/kBT, for which we consider the fixed value b = 0.2. Therefore, the systems are defined by the variables N, and b, and the volume fraction, U, for which we fix the value U = 0.1. At this value, the chains overlap (the overlapping concentrations have been estimated [13] as U* = 0.074 for N = 40 and U* = 0.035 for N = 100). However, overlapping may be weak for the N = 40 chains and

we may observe a behavior closer to the dilute Rouse model in this case. On the other hand, the concentration is relatively close to the previously estimated ODT for these systems [23], specially in the N = 100 case. Each chain consists of two blocks, each composed of units of a different type, A or B. The compositional heterogeneity is introduced by randomly picking out the chain lengths of one of the blocks in the simulation box so that they follow a Gaussian distribution of mean N/2. The standard deviation introduced in this distribution determines the block polydispersity. The number of chains, nc, according to the specifications given above is relatively large, n = 142 for the N = 40 systems and n = 128 for the N = 100 systems. Consequently, the real distributions of block lengths are close to the Gaussian functions used for their generation. This is specifically shown in Fig. 1. Since the AA and BB interactions are not included, the empty sites in the lattice distribute uniformly in the systems, effectively acting as a good solvent. The total energy of a configuration complying with the self-avoiding condition is the sum of the intramolecular and intermolecular 0.06

0.05

Fraction of Chains

self-avoiding chains), but incorporating compositional heterogeneity [24]. We obtained three modes simply correlated with the contributions of heterogeneity, internal and fast Rouse motions. The collective scattering analysis of these simulations was not, however, carried out in detail since the decomposition into modes could not be adequately performed due to the statistical noise of the functions. In the present article, we perform simulations with the bond fluctuation model for symmetric diblock copolymers in a non-selective solvent, including both segregation effects (repulsions between units of different block types) and some amount of compositional heterogeneity. We directly analyze the collective light scattering and obtain the different mode contributions. The results are compared with those of our previous simulations and also with the theoretical predictions and experimental data.

N=40 p=1.10 0.04

0.03

0.02

0.01

0.00 0

10

5

15

20

25

30

35

40

Block Length 0.04

N=100 p=1.10

0.03

Fraction of Chains

5062

0.02

0.01

0.00 0

20

40

60

80

100

Block Length

Fig. 1. Distributions of the number of beads in block A in the polydisperse samples, (solid line) compared with the Gaussian function. Error bars of the distributions are ±0.01 for (a) and ±0.005 for (b).

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i;j

+

 expfiq  ½Ri ðsÞ  Rj ðs þ tÞg

:

ð1Þ

s

The vectors Rk(t) refer to the positions of the different nS = L3 sites within the system and fk(t) is the contrast factor, related with the difference between the scattering factor due to the particular occupation in the site at a given time t and the mean scattering factor of the system. Thus, for the ‘zero-average contrast conditions’ systems, the solvent scattering factor is equal to the system’s mean. Therefore, for symmetric chains, one can set fi = 1 if site i is occupied by unit A, fi = 1 if site i is occupied by unit B, or, finally, fi = 0 if site i is vacant or blocked. Only a discrete number of values of the coordinates of q can be used to apply in this equation, since their values are conditioned by the box size [26]. 3. Results Table 1 contains a summary of the results for hS2i, D and s1 obtained for the two chain lengths. Error bars are

estimated from results directly calculated from several (usually 4) independent statistical runs. (They are not simply obtained from the fits or statistical deviations of a single run, which typically give smaller apparent uncertainties.) The samples with different block polydispersity give results within these error bars. The scattering curves have been fitted to several types of multiexponential functions. Typically, we have found a good fit with two or three exponentials. Based in the decay rates and their intensities (preexponential factors) we have classified the resulting contributions as a ‘fast’ mode, smaller in intensity, which is exhibited by most of the cases and a predominant ‘slower’ mode, which splits into two independent exponentials in some of the cases. In Fig. 2 we present the total intensities (sum of preexponentials or fitted value of the curves at t = 0) corresponding to a chain of N = 40 obtained from two simulations performed with different amounts of the block polydispersity p. This quantity is defined as the ratio between the weight and number averages of the block chain lengths present in the simulation sample. We set p = 1.02 and p = 1.10 for these systems, with the same values for the polymer volume fraction and energetic interaction parameter previously mentioned. Since the scattering results have to be calculated with some discrete values of q of defined orientations, they are sensitive to the particular orientation of the system with respect to the different directions of the q vectors. Actually, the data obtained from the fits of scattering functions corresponding to individual q vectors are treated as a statistical sample in order to obtain the reported means and error bars. In Fig. 3 we present the results obtained for decay rates of the ‘slower’ mode for the same cases. The slower mode is assumed to contain both the internal and heterogeneity contributions, as a single term or separated into two terms that we assign to the theoretical slow and fast modes.

1.2

1.0

Total Intensity, S(q)

energies corresponding to all the pairs of interacting units. The configuration is accepted or rejected by comparing this ‘new’ energy with the energy of the previous (or ‘old’) configuration, according to the Metropolis criterion. The initial configurations are taken from previously equilibrated samples. (These equilibration runs were performed with the more efficient ‘reptation’ moves, where a bead is removed from a randomly chosen end of a chain and a new bead is inserted in a random position at the opposite end [22].) Typical samples consist of 106 attempted moves per chain for equilibration, followed by 106 time units for the statistical production runs. Trajectory coordinates are saved at intervals of 100 time units. The properties that we have calculated from these trajectories are: (a) the chain’s mean size, represented by the quadratic mean radius of gyration, hS2i, (b) the diffusion coefficient obtained from the mean square displacement of the center of masses of the chains, D, (c) the first Rouse relaxation time, s1, obtained by computing the time-correlation function of the first Rouse mode and fitting its decay to a single exponential and (d) the dynamic collective scattering function of the system (or simulation box), that is calculated as * nS X 1 Sðq; tÞ ¼ 8nS fi ðsÞfj ðs þ tÞ

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0.8

0.6

0.4

0.2

Table 1 Radius of gyration, diffusion coefficient and Rouse relaxation time values for the different symmetric diblock copolymer chains with U = 0.1, b = 0.2 N

hS2i

104D

104s1

6hS2i/Ds1

40 100

88 ± 2 244 ± 6

4.7 ± 0.3 1.7 ± 0.2

4.9 ± 0.3 31 ± 3

23 ± 3 28 ± 6

Properties are expressed consistently with the model basic units as explained in the text.

0.0 0.00

0.01

0.02

0.03

0.04

0.05 2

0.06

0.07

Scattering Variable, q

Fig. 2. Total intensities for the N = 40 copolymer. The open symbols correspond to the case p = 1.02 and the filled circles to the case p = 1.10. The solid curve is the prediction for independent Gaussian copolymers, Eq. (2).

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6

5

Total Intensity, S(q)

5

Mode Decay Rates, 10 Γi

8

6

4

4

3

2

2

1

0 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 0.00

0.07

0.01

2

6

0.04

1.4

1.2 4

1.0

5

Mode Decay Rates,10 Γi

5

0.03 2

Fig. 4. Total intensities for the N = 100 copolymer, p = 1.10 (symbols). The solid curve is the prediction for independent Gaussian copolymers, Eq. (2).

8

Mode Decay Rates, 10 Γi

0.02

Scattering Variable, q

Scattering Variable, q

2

0 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

2

Scattering Variable, q

Fig. 3. Decay rates for the slower mode of the N = 40 copolymer. The open symbols correspond to the faster contribution. The filled symbols are for the slower contribution. The mixed symbols correspond to the modes that could not be separated into two contributions. Lines are the basic predictions for an internal or Rouse (dashed) and a heterogeneity (solid) contribution, see the text. (a) p = 1.02, (b) p = 1.10.

In Fig. 4, we show the total intensities for the case N = 100, p = 1.10, with the same volume fraction and energetic parameter. Fig. 5 shows the decay rates of the ‘slower’ mode observed for this system. The mode is also separated, when possible, into two contributions. 4. Discussion The results in Table 1 are consistent with previous data obtained for the same model of diblock copolymers at different concentrations (see Table 1 of Ref. [22]). The combination of the three quantities into the single parameter 6hS2i/Ds1 should give a nearly constant quantity, if the system friction affects the three properties similarly [27]. Our results are close to the value 3p2 ffi 30 that corresponds to single Gaussian chains. The parameter should actually be smaller for chains in a good solvent. The closer approach

0.8

0.6

0.4

0.2

0.0 0.00

0.01

0.02

Scattering Variable, q

0.03

0.04

2

Fig. 5. Decay rates for the slower modes of the N = 100 copolymer, p = 1.10. The open symbols correspond to the faster contribution. The filled symbols are for the slower contribution. The mixed symbols correspond to the modes that could not be separated in two contributions. Lines are the basic predictions for an internal or Rouse (dashed) and a heterogeneity (solid) contribution, see the text for details.

to the Gaussian limit is found for the N = 100 systems, that, at a given volume fraction, should exhibit greater chain overlapping and a more effective screening of excluded volume effects. In any case, the present model, with only repulsions between units belonging to different types of blocks, does not show the stronger deviations found for this parameter in systems with an attractive interaction between similar units [22]. (In these latter systems, the mobility is retarded because of their proximity to the macroscopic phase separation.) The simulation results for the intensities of the N = 40 systems are shown in Fig. 2. They are compared with the theoretical value corresponding to the independent contribution of individual Gaussian copolymer chains

E. Rodrı´guez, J.J. Freire / Journal of Non-Crystalline Solids 352 (2006) 5060–5066

    ex x 3 þ  ; S theor ðxÞ ¼ N U 8=x2 ex=2  4 4 4

ð2Þ

where x = q2hS2i. The simulation for the systems with small block polydispersity, p = 1.02, should give results close to those of a monodisperse sample. However, it can be observed that the two systems show intensity values significantly greater than this simple theoretical prediction. This effect is due to segregation effects, even though these systems are not very close to ODT (according to our previous estimation [23], the ODT for N = 40 and U = 0.1 is expected to occur at b ffi 0.70 for the present model). The higher increase observed for the p = 1.10 system reflects a moderate contribution due to the block polydispersity. In Fig. 3, we present the decay rate values for the slower mode obtained for the N = 40 systems. We have been able to separate this mode into two contributions in some cases. This separation is more easily achieved for the higher polydispersity, p = 1.10, system. In the p = 1.02 system, the separation can actually be performed only for a single value of q. Furthermore, the values for the decay rates of the nonseparated modes are closer to the internal contribution than for p = 1.10. In both cases, the slowest contribution or the non-separated decays only show a faint decrease of the decay rates near the location of the intensity maximum, which, in any case, is within the error bars. Therefore, we cannot perform a precise verification of the theoretically predicted retardation in the slowest mode. It should be mentioned that the fastest mode of smaller intensity, also found from the simulation curves, follows a less systematic variation with q. We attribute this mode to separated contributions of some of the faster internal motions in the chain. A similar explanation was given for the faster mode exhibited by our simulation results for semidilute solutions of monodisperse diblock chains [13]. The increase of the decay rate for the slower mode at the highest value of q should be also caused by the contribution of shorter internal chain motions that cannot segregate to the faster mode, showing some deviation from the approximated Rouse prediction, which is also noticed for simple diblock copolymer chains [8,13]. In Fig. 4, we observe that the intensities obtained for the N = 100 case are significantly higher with respect to the theoretical curve than for the N = 40 systems. This further increase is due to the closer proximity of the ODT. This system is significantly closer to the ODT since, according to our estimations [23], the transition for N = 100 and U = 0.1 occurs at b ffi 0.3 for the present model. This proximity also causes a significant anisotropy in the model, which is the main cause of the oscillations and higher error bars of the intensity results. (The oscillations should disappear if the average could be performed over all the orientations of the q vectors.) Fig. 5 shows that a clearer split of the ‘slower mode’ for N = 100 into two contributions has been observed for most of the points. Also in this case, the two contributions are close to the simplest predictions for the ‘heterogeneity’

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and ‘internal’ contributions. A more elaborate analysis of these contributions, in terms of the slow and fast theoretical mode predicted by the RPA is not undertaken in the present case, due the intrinsic uncertainties associated to these simulations. Recent experimental results for diblock copolymers of ultrahigh molecular weight actually show two modes with decay rates approximately following parallel straight lines, in good qualitative agreement with this simple behavior, see Fig. 6 of Ref. [28]. Similarly to the N = 40 case, the simulation data do not show any clear retardation of the slower mode in the proximity of the intensity maximum. Several sets of recent experimental data point out that this effect is smaller than the increase in intensity for most systems [11,28]. Also as in the systems with N = 40, the influence of shorter internal motions of the chains is manifested by the increase of the decay rate for the highest value of q and also by the presence of a fast mode of smaller intensity. As previously stated, the present Monte Carlo simulations cannot describe hydrodynamic interaction effects. These effects do not seem to provide significant contributions in the ODT location for melts of long chains [29] but are certainly present in semidilute but weakly overlapped systems closer to the dilute regime, as in the case of our N = 40 chains. The main influence should be incorporated through the diffusion coefficients and the relaxation times which, actually, exhibit different dependencies with the chain length when hydrodynamic interactions are taken into account. However, this influence can be easily taken into account if the values of these magnitudes incorporated into the discussion have been consistently obtained from the same simulations. However, the influence of the faster chain motions inside the reptation tube may give also give a contribution for intermediate and high values of q, even for more overlapped systems. The theoretical description based in the RPA without fluctuations [11] assumes that the motion of a polymer chain is a combination of reptation and internal motions, but the latter are neglected or described through the Rouse description (without hydrodynamic interactions). The comparison with experimental data of semidilute solutions show important differences (a factor of about 4) in intensities and rates and part of these differences may be related with the presence of hydrodynamic interactions in the experimental systems. On the other hand, our previous study of the dynamic scattering of single chains obtained from the Rouse–Zimm scheme [8], also lead to a qualitative description in terms of the diffusion coefficient and relaxation times where the hydrodynamic interactions should be taken into account in the evaluations of these magnitudes. The qualitative description in terms of different exponentials (with the main contribution decaying with the rate Cint = 1/s1 + Dq2) is, however, maintained, though the relative intensities of the resulting exponentials may be quantitatively different (they are affected through the change in the Rouse–Zimm normal modes, weakly dependent on the hydrodynamic interaction strength).

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Certainly, dynamic simulations including both hydrodynamic interactions and entanglement effects would be more adequate to avoid the problems addressed in the previous paragraphs. However, we believe that the present type of Monte Carlo simulations are able to capture the qualitative description of the systems, particularly the description and the q-dependence of the most significant modes in terms of the D and s1. The effect of hydrodynamic interactions would be mainly manifested in the quantitative values and relative intensities between the modes. 5. Conclusions The present simulations were carried out using the bond fluctuation model representation of diblock copolymers with block polydispersity They are able to give an acceptable description of the dynamic scattering modes predicted theoretically and also shown by some experimental samples. The total intensities increase with the amount of polydispersity and, more significantly, as the systems approach the ODT. The location of the main mode (decay rates) and its variation with q can be more easily decomposed into two contributions for the systems where these two features are enhanced. Moreover, the resulting values are close to the simplest predictions for the heterogeneity and internal contributions. The retardation of the decay rates of the slowest mode in the proximity of the intensity maximum and the more subtle behavior due to ‘eigenvalue effect’ predicted by the more complex RPA theory cannot be confirmed by the present simulation data. They would require a more exhaustive investigation focused on very particular systems and employing large simulation boxes to increase the number of useful values of q. Our simulation data also show a faster mode, which is also detected in simulations with monodisperse samples, and it is probably due to contributions of shorter internal motions of individual chains. The shorter internal motions that do not segregate in this fast mode are manifested by an increase in the decay rate when compared to the basic Rouse prediction at the highest q values. This work presents the first detailed analysis of the collective scattering function of copolymer systems with block polydispersity obtained through numerical simulation. In spite of their large uncertainties, the proximity of these results to the theoretical expectations shows that simulation techniques using coarse-grained (lattice) models can be used effectively as a promising route to understand the complex dynamic behavior of this type of data.

Acknowledgements This work has been supported by Grant BQU200204626-C02-02 of the DGI (MCYT, Spain). E.R. is thankful for a FPI Fellowship associated with this Project. We thank Dr C. McBride (IQF Rocasolano, CSIC; Madrid, Spain) for reading and commenting on the manuscript. References [1] J.W. Hamley, The Physics of Block Copolymers, Oxford University, Oxford, 1998. [2] M.W. Matsen, F.S. Bates, Macromolecules 29 (1996) 1091. [3] G.H. Fredrickson, E. Helfand, J. Chem. Phys. 87 (1987) 697. [4] A. Rebei, J. De Pablo, Phys. Rev. E 63 (2001) 041802. [5] G.H. Fredrickson, L. Leibler, Macromolecules 22 (1989) 1238. [6] W. Brown (Ed.), Dynamic Light Scattering, Clarendon, Oxford, 1993. [7] R. Sigel, S. Pispas, N. Hadjichristidis, D. Vlassopoulos, G. Fytas, Macromolecules 32 (1999) 8447. [8] A. Rey, J.J. Freire, J. Chem. Phys. 101 (1994) 2455. [9] A.N. Semenov, S.H. Anastasiadis, N. Boudenne, G. Fytas, M. Xenidou, N. Hadjichristidis, Macromolecules 30 (1997) 6280. [10] T. Jian, S.H. Anastasiadis, A.N. Semenov, G. Fytas, K. Adachi, T. Kotaka, Macromolecules 27 (1994) 4762. [11] K. Chrissopoulou, V.A. Pryamitsyn, S.H. Anastasiadis, G. Fytas, A.N. Semenov, M. Xenidou, N. Hadjichristidis, Macromolecules 34 (2001) 2156. [12] Z. Liu, C. Pan, T.P. Lodge, P. Stepanek, Macromolecules 28 (1995) 3221. [13] A.M. Rubio, J.F.M. Lodge, J.J. Freire, Macromolecules 35 (2002) 5295. [14] M. Murat, G.S. Grest, K. Kremer, Macromolecules 32 (1999) 595. [15] A.J. Schultz, C.K. Hall, J. Genzer, J. Chem. Phys. 117 (2002) 10329. [16] R.D. Groot, T.J. Madden, J. Chem. Phys. 108 (1998) 8713. [17] S.Y. Qi, Z.G. Wang, Phys. Rev. E 55 (1997) 1682. [18] K. Binder, M. Mu¨ller, Curr. Opin. Colloid Interface Sci. 5 (2000) 315. [19] Q. Wang, P.F. Nealey, J.J. de Pablo, Macromolecules 35 (2002) 9563. [20] I. Carmesin, K. Kremer, Macromolecules 21 (1988) 2819. [21] K. Binder (Ed.), Monte Carlo and Molecular Dynamics Simulations in Polymer Science, Oxford University, New York, 1995. [22] A.M. Rubio, M. Storey, J.F.M. Lodge, J.J. Freire, Macromol. Theor. Simul. 11 (2002) 171. [23] J.J. Freire, C. McBride, Macromol. Theor. Simul. 12 (2003) 237. [24] L.A. Molina, J.J. Freire, J. Chem. Phys. 109 (1998) 2904. [25] M. Wittkopp, S. Kreitmeier, D. Go¨ritz, J. Chem. Phys. 104 (1996) 3373. [26] A. Sariban, K. Binder, Colloid Polym. Sci. 267 (1989) 469. [27] J.J. Freire, L.A. Molina, A. Rey, K. Adachi, Macromol. Theory Simul. 8 (1999) 321. [28] P. Holmqvist, G. Fytas, S. Pispas, N. Hadjichristidis, H. Tanaka, T. Hashimoto, Macromolecules 37 (2004) 4909. [29] M.A. Horsch, Z. Zhang, C.R. Iacovella, S.C. Glotzer, J. Chem. Phys. 121 (2004) 11455.