ВЫСОКОМОЛЕКУЛЯРНЫЕ СОЕДИНЕНИЯ, Серия C, 2013, том 55, № 7, с. 902-910

УДК 541.64:539.2


© 2013 г. Claudine Gross and Wolfgang Paul

Institute of Physics, Martin-Luther-University, 06099 Halle, Germany e-mail: Wolfgang.Paul@Physik.Uni-Halle.De

Abstract—We present results for a recently introduced soft-particle type model for diblock copolymers. Our focus will be the interaction of this model with confining walls and the possibility to direct the microphase morphologies by tailoring the interactions with these walls. We begin by presenting its bulk phase diagram and our method for determination of the different phases. We interpret the phase behavior in comparison to experimental data as well as other model results. By a systematic coarse-graining of chemically realistic simulations, one can obtain the effective potential acting between the walls and the repeat units of our soft qua-drumer model. We employ this coarse-grained potential then for simulations of the confined case for several strengths of attraction to the walls and demonstrate when extending the film thickness leads to nucleation of a new lamella in the center of the film and when it leads to reorientation transitions of the lamellar microphase.

DOI: 10.7868/S0507547513050073


Micro-phase-separation occurring in diblock copolymer melts has attracted interest since a long time [1—3]. From the point of view of basic science it is interesting that the transition occurs into an ordered structure characterized by a definite finite length scale in the nanometer regime [4]. From the point ofview of applications, the self-assembled nano-structured morphologies can serve as templates for organization of other materials with applications for instance in the design of nano-composite materials [5] or organic optoelectronic devices [6]. Furthermore, directed self-assembly of copolymers at structured interfaces is an important tool in nano-lithographic processes employed for chip design [7] and a theoretical prediction of these self-assembly processes is highly desirable [8, 9].

The most successful method for the description of the phase behavior of block-copolymers has been self-consistent field theory (SCFT) [10—15]. However, SCFT is a mean-field approach ignoring fluctuations which qualitatively change the phase diagram [16— 20]. Several approaches have been pursued in the last decade to improve the field-theoretic descriptions to include fluctuations [13, 14, 21] or to include conformational fluctuations of the chains in a hybrid SCFT-molecular approach [22]. The first order nature of the transition from the disordered to the lamellar state [23], as well as direct first-order transitions from the disordered to cylinder and gyroid (or perforated lamella) phases [12], for example, are not predicted within standard SCFT These differences can be reproduced by simulations using coarse-grained molecular models (see for example [24—31]), however, these

models demand very time-consuming simulations. Some groups have therefore employed models on the blob-scale [32, 33], employing dissipative particle dynamics to improve efficiency.

All structures generated within the micro phaseseparation are on the scale of the radius of gyration of the chains. Arguably, this should enable a modeling approach working with effective potentials on that same scale. This idea has motivated simulations employing soft-particle models of block copolymers [36— 38]. These models incorporate effective interactions on the scale of the radius of gyration of the chains and they capture fluctuations on that scale, too, which arguably are the relevant ones for the determination of the topology of the block copolymer phase diagram. In the model we suggested [38] the block copolymer chains are modeled as a soft quadrumer to incorporate the non-spherical shape of polymer melt chains. We will introduce this model and our simulation and analysis technique in section Model. Section Results for Bulk will then discuss some results on the bulk phase behavior which has been published earlier [38, 39]. In section Coarse-Graining Procedure we will present the coarse-graining procedure to obtain the wall potential for our soft quadrumer model while section Results for the Confined Lamellar Phase discusses our results for the phase diagram of confined model systems. Finally, in section Conclusions and Outlook we will present a summary and outlook for future work.


The idea to represent polymer chains in solutions or melts by effective soft particles has found much in-

terest in the last years [40] and quite some knowledge has been obtained on the effective interactions in such soft particle models in solutions and in the melt [41— 43]. Using molecular simulations to obtain pair-correlation functions between mass centers of different chains in solution and employing liquid state theory to invert these to obtain effective interactions between the chains, it has been found that the chains interact by soft potentials with an amplitude at separation zero of about 2k BT — relatively independent of concentration — and that the interaction decays to zero for a distance between one and two Rg (radius of gyration). These findings are the starting point for building our model in which we represent a Gaussian chain by two two soft particles interacting with entropic repulsive potentials with other chains as well as with one another. The two chain halves are bound to each other by en-tropic bonds. Therefore, all interactions between like segments are proportional to temperature. When we join two such chains — the A-block and the B-block — we obtain a diblock copolymer made of four soft particles, the soft-quadrumer model. For the interactions between unlike particles we employ the same form of potential as for the like particles but choose the interaction strength as a constant. Reducing temperature then leads to an effective repulsion between unlike soft particles inducing the micro phase separation [38]. The non-bonded interactions are given by

Unb =

^ aa

Uab =

1 - 3| + 21 r


0 r > —

2 1 -31 J— | + 2|-r


r <-r,

- AB 0 r > - AB


r < -


Here uAb = 1/2(ct^ + aBB) and the aaa are the diameters of spheres representing half the blocks. For the bonded beads we add an entropic spring potential

TTbond Tjnb , 1 r

Uaa = Uaa + 1-6T I

TTbond Trnb * srji [ r U AB = U AB + 1.6T I



The control parameters of the phase diagram are composition, f, and incompatibility, %N, where N is the chain length. We will assume equal statistical segment length for both blocks. Then we have (with N A

being the length of block A and NB the length of block B)

Na _

f =

Na + Nb

bA(NA /2)

bA(Na/2) + bB(NB /2)




2 2 a a +a b

The condition of fixed chain length N translates into the requirement

2,2 ~ 2 a A + a B = 2a ,

2V , (4)

where aA = aB = a is the size of the spheres at symmetric composition f = 0.5. We choose a = 1 as our length unit. A composition f then translates into the following diameters of the two types of soft spheres:

a a =V2f and Gb =V 2(1 - f ). (5)

The asymmetry of the non-bonded interactions is given by

As - s AB - ~(s AA + SBB)-


The following is the natural definition of %N in our quadrumer model

XN = 4 ^ = 4(2-2T) = 8 ( I _ 1 T T \T


We performed Monte Carlo simulations in the canonical ensemble to determine the bulk phase diagram as well as the behavior in confinement. To identify phase morphologies we used a newly developed algorithm [38] built upon identifying clusters of like particles and determining their shape (through the eigenvalues of the gyration tensor of the clusters) as well as the cluster size distribution. A continuum variant [44] of the Hoshen-Kopelmann [45] algorithm was used for cluster identification.


Results for the Bulk

Microscopic imaging has been a long-standing tool of experimental identification of diblock copolymer morphologies. The reason is that all of them, lamella, cylinder, spherical, and bi-continuous have unique signatures in the combination of shape, size and relative number of the phase segregated clusters. This qualitative finding is what we quantify by cluster number distributions and gyration tensor eigenvalues. A lamellar phase, for example, is described by equal cluster size distributions for A and B species, peaking at about the same number of clusters and furthermore by equal average values for the eigenvalues of the gyration tensor. We can assume the lamella to be oriented more or less parallel to the boundary of the simulation box (a mismatch between lamella spacing and size of the simulation box results in a slight tilt of the lamellae). Two of these are then of the order of the simula-

Fig. 1. (a) Configuration snapshot of the lamellar phase at pc = 1.3,/= 0.52 and %N = 27. (b) Cluster number distributions for majority (A) and minority (B) phases for the same parameters. 1 — A clusters, 2 — B clusters.

tion box size and the other of the order of the lamella spacing.

As an example, let us look at the cluster number distribution. For a lamellar phase we are showing a configuration snapshot in Fig. 1a and the corresponding cluster number distribution in Fig. 1b. Obviously,








Disordered X Lamellar * Cylindrical □ Perforated lamellar

0.9 f

Fig. 2. Region of the phase diagram for pc = 1.3 containing the lamella, the perforated lamella, the cylinder and the disordered phase. The phase diagram is, of course, symmetric with respect to f = 0.5.

both components consist of about the same number of clusters. When we look at the distribution of eigenvalues for the gyration tensor of the clusters we have identified, we find for both components two peaks close to the linear size of the simulation box and one peak corresponding to the thickness of t

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