自洽场理论Fourier空间解法：ABC星型与线型嵌段共聚物相图计算
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摘要
软物质的自组织已经成为凝聚态物理领域的一个研究热点。预测、发现和理解软物质自组装的新的有序结构及研究软物质自组织过程中的动力学等问题将为我们深入理解分子组装的物理机制以及进行新材料的结构设计作出贡献。嵌段共聚物作为研究软物质自组织现象的一个主要对象,在本体和溶液体系中,可以形成多种有趣的有序结构。现在已知的是,AB两嵌段高分子本体可以形成四种微相分离结构：交替层状结构(L)、六角排列柱状相(C)、体心立方球状结构(S)和双连续Gyroid结构(G)。Matsen于1994年采用自洽场方法计算得到了AB嵌段高分子的相图。最近,在Gyroid相区附近又发现了一种具有Fddd空间群的O70结构。
当前,自洽平均场理论已经成为研究高分子体系相平衡态结构问题十分重要的理论方法。该方法最初由英国理论物理学家Edwards于上世纪60年代中期提出,随后由Helfand, Noolandi等将该理论引入到了嵌段共聚物等多相高分子体系中。过去十几年来,对探索自洽场理论数值解法的努力大大推进了该理论在高分子科学领域中的应用。目前发展成熟的自洽场理论数值解法包括：Matsen-Schick谱方法、Fredrickson-Drolet实空间解法和Rasmussen等提出的准谱方法,它们分别提出于上世纪九十年代中期、末期和本世纪初。这些方法各有优缺点。Matsen-Schick谱方法中,空间变量函数以某些具有设定空间群对称性的基函数展开,进而在Fourier空间求解自洽场方程。该方法适用于计算所设定结构对称性的形态的自由能,在计算体系相图方面优势显著,但不适用于新形态的预测。Drolet-Fredrickson实空间方法和准谱方法等虽然可以预测嵌段共聚物的新结构,但是计算大尺寸体系时较难得到十分规整有序的结构。而且,对那些复杂双连续型网络结构(如Gyroid、金刚石结构、Fddd等)的搜索不仅十分困难,在确定这些复杂结构的对称性空间群方面更有所欠缺。最近,鉴于在线型嵌段共聚物中发现的有序结构多为中心对称结构,Guo等提出了一种可用于搜索中心对称结构的具有一定“普适性”的谱方法。在该方法中,所有空间变量函数以余弦函数展开,进而在Fourier空间进行求解自洽场方程。然而,在嵌段共聚物某些体系中,如星型共聚物体系、嵌段共聚物共混体系等,会有出现非中心对称结构。因此,对具有非中心对称结构的嵌段共聚物体系的新结构预测方面Guo等发展的方法将不再适用,需要继续发展新的方法。在此背景下,本论文主要目的是发展一种具有真正普适性的方法：不仅能搜索嵌段高分子体系所有可能出现的结构(包括中心对称结构和非中心对称结构),又能够从计算结果中分析所得结构的对称性方面的信息,从而方便地确定一些复杂结构的对称空间群。将这些结构作为备选态采用传统的谱方法进行自由能的精确计算,有望可以得到体系的精确相图。为此,我们进一步拓展了Guo等的谱方法。其基本思想是：基于嵌段共聚物可以形成规则的有序周期性结构,我们可以将与空间有关的所有变量进行Fourier级数(包括余弦函数和正弦函数)展开,使自洽场理论方程均在Fourier空间求解。该方法可以近似看作：起初我们给定一个具有所有空间群对称性的“基函数库”,随着整个计算迭代过程进行,会自动筛选出与设定体系分子参数相匹配的那些具有某些对称性的基函数。从而实现搜索新结构的功能。本方法的缺点是,由于空间变量的Fourier展开项个数受限于计算能力的制约,在计算过程中,只能取有限个基函数展开。这将会给计算结果的准确性带来影响。
方法建立后,我们对其可靠性和有效性进行了检验。首先,采用此方法考察了最简单的AB两嵌段高分子体系。我们成功得计算得到了AB嵌段共聚物本体体系中已知的四种结构：层状结构、六角柱状结构、体心立方球状结构和Gyroid双连续结构。然后,我们将该方法用于ABC三嵌段高分子的形态的预测。同样,我们成功的预测出了线型高分子体系中已经实验观察到和理论预测出来的一些典型结构,如交替型结构、核壳型结构、复合型结构等等。最后,对ABC星型高分子体系的结构预测做了考察。计算得到了该体系中典型的几种二维柱状多边形结构：[6.6.6],[8.8.4],[8.6.4；8.6.6],[10.6.4；10.6.6],[12.6.4；8.6.4],[12.6.4]等。综合对以上几个体系的检验,我们得出结论：本论文中发展的自洽场理论Fourier空间解法具备嵌段高分子体系中的结构预测功能,可以被用来研究和搜索嵌段共聚物中的新的有序相结构。
在嵌段共聚物相分离实验中,X射线衍射方法是测定其形成的有序结构的重要方法。根据散射理论,我们推导出了以Fourier空间数值解的具体形式表达的体系散射函数。以此为基础,分别讨论了AB两嵌段高分子、ABC线型和星型嵌段高分子体系中已知的有序结构的散射性质,得到了这些结构的散射强度与散射波矢模的关系图谱,将这些计算结果同实验结果进行了对比,在这些有序结构(包括已知的那些复杂性网络结构)的散射特征方面理论计算和实验结果取得了十分一致的结论。这些结果表明：该方法具备确定有序态结构的能力,包括对于那些复杂的双连续网络型结构的对称性的确定,从而使得该方法在预测复杂嵌段共聚物相形态方面具有各大的优势。
ABC星型嵌段共聚物由于其特殊的链拓扑结构,赋予了它们十分独特的自组装行为。鉴于目前对该体系的本体组装行为理解还不是十分深入,为了较好的认识该类型嵌段高分子的本体中的组装形态,我们采用前面提出的自洽场理论Fourier空间解法对ABC星型嵌段共聚物体系的本体平衡态相分离形态做了详细的考察。为了提高计算效率,并考虑到ABC星型高分子在三个支链链长相当时多形成二维柱状多边形结构,所以我们将自洽场计算限定在二维空间中。我们主要考察了两类星型共聚物体系。第一类为三组分具有相同的相互作用,我们称之为对称相互作用体系,χABN=χBCN=χACN=30.0,第二类则为非对称相互作用体系,χABN=χBCN=25.0,χACN=37.0。我们分别计算得到了这两类体系的三元相图。在对称相互作用体系中,计算得到了六种二维柱状多边形结构,包括：[6.6.6]、[8.6.4；8.6.6]、[8.8.4]、[10.6.4；10.6.6]、[12.6.4；8.6.4]和[12.6.4]；将该计算结果同Monte Carlo模拟、DPD模拟等方法的结果进行了比较,得到了一致的结果。在第二类体系中,则只得到了四种有序结构：[8.6.4；8.6.6]、[8.8.4]、[10.6.4；10.6.6]和[12.6.4]。我们将第二类体系计算结果同实验中ISP体系发现的结构做了对比,发现在相转变规律上得到了一致的定性结论,但在定量的相转变点上存在一些不一致性。这种不一致性可能是由于自洽场计算中所设定的分子参数等同实验实际体系不一致的原因。
相比于AB两嵌段共聚物体系来说,ABC三嵌段高分子体系具有更多可调参数,这就赋予了该体系更加丰富且更难穷尽的有序结构。在ABC线型嵌段高分子本体相行为方面,实验和理论都做了一些有意义的工作。实验中主要详细考察了两个体系：ISO和SBM (SEBM)体系,理论计算等方面目前已得到的两个相图也均是对应于这两类体系。这些实验和理论研究工作均是给出了固定链段间相互作用参数情况下,不同的组分比例而对应的平衡态相结构的信息。为了从另外一个侧面对ABC三嵌段高分子体系本体相行为作一些深入了解,即固定组分比例而改变链段间的相互作用参数,考察体系的有序结构将怎样变化,我们主要考察了对称体系中,即χABN=χBCN,fA=fC,相互作用参数χABN和χACN的变化对嵌段高分子的平衡态结构的影响。为了提高数值计算稳定性和计算效率,我们将体系限定在弱分凝至中等强度分凝情况下进行考察。我们探讨了三个不同组分体积分数时的情况,分别是：fA=fC=0.201、fA=fC=0.25和fA=fC=0.18。我们发现,在不同的组分相互作用情况下,第一个体系可以形成多种不同的有序结构,包括交替型球状相、交替型柱状相、交替型Gyroid、交替型金刚石结构和Gyroid结构。对于第二类体系,通过改变相互作用参数,我们得到了交替型Gyroid和三层层状结构两种形态。第三类体系计算结果表明,在不同的组分相互作用参数下,体系可以发生从层+球结构向三层层状相的转变。我们分别对这些相转变过程的物理机制做了初步的探讨。
综上所述,本论文发展了自洽场理论Fourier空间解法,本方法同时具备预测新结构和确定所得结构对称性信息的功能。应用该方法我们分别研究了ABC星型嵌段共聚物和ABC线型嵌段共聚物的本体相行为,计算得到了它们的相图。
Self-organization of soft condensed matters has been one of the hot topics in the field of condensed matter physics. Full understanding of their structural formations, along with their dynamics features during phase transitions will make contributions to design novel functional materials. Soft matters include polymers, colloids, surfactant, biomacromolecules etc. As a typical soft matter, block copolymers have taken considerable attentions for decades due to their fruitful nano-scale ordered structures and potential applications. It has been well known that there are five equilibrium ordered phases in block copolymer melts, alternating lamellae, hexagonally-packed cylinders, center-body cubic spheres, double gyroid and Fddd (O70).
Self-consistent field theory (SCFT) has been one of the most important and successful theories in polymer science. The SCFT has its origin in the work of Edwards in 1960s. Later, this theory was adapted explicitly to treat block copolymers by Helfand and Noolandi etc. During last decades, establishment of various numerical methods for solving SCFT equations, which include "spectral method" proposed by Matsen and Schick in 1994, "real-space method" proposed by Drolet and Fredrickson in 1999 and "pseudo-spectral method" proposed by Rasmussen etc. in 2001, have greatly improved the applications of SCFT in the field of polymer science. These methods have their advantages and disadvantages. Take "real-space method" for instance, although this method is good at searching new ordered structures in block copolymers, difficulties are encountered at reaching long-range ordered structures and determination of space group of complex ordered phases, especially those of complex networks, like gyroid and Fddd. For Matsen-Schick spectral method, all spatially varying functions are expanded in terms of basis functions with priori symmetry and SCFT equations are cast in Fourier space. Consequently, the spectral method is only good at computation of phase diagrams. Recently, Guo et al proposed a generic method for solving SCFT equations, in which all spatially varying functions are expanded in terms of cosine functions, provided that most of ordered phases observed experimentally are centrosymmetric. There are, however, some non-centrosymmetric structures in block copolymer melts, especially in star-shaped terpolymers, where the generic method for SCFT is no longer valid. Under this situation, we aim at proposing a generic approach to solution of self-consistent field theory (SCFT) equations for block copolymers, which combines the capabilities of searching new ordered phases (including centrosymmetric and non-centrosymmetric structures) and determination of space groups of obtained structures together. In this method, all spatially varying functions are expanded in terms of Fourier series (including cosine and sine functions) which are essential determined by computational box parameters. Then, SCFT equations can be cast in terms of expansion coefficients. This method can be looked as an expansion of Matsen-Schick spectral method. The advantage of the approach consists in fact that structural symmetries of resulting ordered phases can be easily deduced from expansion coefficients of nonzero values, which will be clearly demonstrated for complex phases in block copolymers.
To evaluate the capabilities of generic Fourier-space method for SCFT, we firstly applied this method to AB diblock copolymer melts. As expected, we successfully obtained the ordered phases including alternating lamellae, hexagonally-packed cylinders, body-centered cubic spheres and double gyroid. Then, the equilibrium ordered phases in ABC linear triblock copolymers melts are computed using the generic Fourier method proposed. Some typical phases, including alternating type, core-shell type and the decorated type structures, are obtained successfully. For some non-centrosymmetric structures in ABC star-shaped terpolymers, we extend the Fourier method to the equilibrium phases in ABC star-shaped terpolymers. Some two-dimensional cylindrical structures are computed, among which are [6.6.6], [8.8.4], [8.6.4;8.6.6], [10.6.4;10.6.6], [12.6.4;8.6.4], [12.6.4]. Note that [6.6.6] and [10.6.4; 10.6.6] are two typical structures of non-centrosymmetric space group. With the above verifications, a conclusion is reached that the generic Fourier method for SCFT is capable of searching new ordered structures with any space groups.
Structural determinations of ordered phases in experiments are made via small-angle X-ray scattering (SAXS) method. We have mentioned that the Fourier method proposed aims at combination of capabilities of searching new structures and of determination of space-groups of obtained ordered phases. In order to make a close comparison with SAXS results in experiments, derivations of the scattering intensities of ordered phases in terms of Fourier coefficients in our method is made. Then, the scattering intensities of some typical ordered morphologies in AB diblock copolymers and ABC triblock copolymers are computed and compared with those in experiments. A good consistence between the theoretical scattering functions and those in experiments proves the capability of determination of space groups of ordered structures in block copolymers.
ABC star-shaped terpolymers have attracted attentions for years due to their fruitful ordered morphologies. The most distinction of phase behaviors of star terpolymers from that of linear triblock copolymers lies in fact that junction points in ABC star terpolymers be arranged along one-dimensional lines resulting from topological constraint, while in linear ABC triblock copolymers two-dimensional plane could be allowed for the connecting points between neighboring blocks. The spatial arrangement of junction point in star-shaped terpolymers leads to two-dimensional cylinder-type morphologies under assumption that three polymer chains are totally incompatible and long enough. With the generic Fourier method for SCFT, the equilibrium phases of ABC star-shaped terpolymers have been studies. Two broad types are investigated in detail:one with symmetric interactions,χABN=χBCN=χACN=30.0, and one with asymmetric interactions,χABN=χBCN=25.0,χACN=37.0, corresponding to ISP star-shaped terpolymers. The triangle phase diagrams are obtained. For the former, six ordered morphologies are obtained, including [6.6.6], [8.6.4;8.6.6], [8.8.4], [10.6.4; 10.6.6], [12.6.4;8.6.4] and [12.6.4], which is in consistence with those simulations by Monte Carlo and DPD. For asymmetric interactions, two series of star terpolymers are studied, A1.0B1.0Cx, A1.0B1.8Cx. After comparison with experimental results for ISP, a qualitative agreement is reached, while there are few quantitative agreements. These discrepancies maybe due to the inconsistence of molecular parameters used in SCFT with those in real star terpolymers, which are hardly determined experimentally for now, like Flory-Huggins interaction parameter, statistical segment length, etc.
ABC linear triblock copolymers reveal fruitful phase behaviors in comparison with AB diblock copolymers, due to their vast parameter space. Thanks to efforts of experimental and theoretical workers on block copolymers, some knowledge has been collected on equilibrium phase behaviors of ABC linear triblock copolymers. Only few samples of ABC linear triblock copolymers, however, have been studied in detail, including ISO and SBM (SEBM). The phase diagrams that have been computed theoretically are for these two types of triblock copolymers, which provide the information of equilibrium phases under each molecular parameter. For obtaining another insight into phase behaviors of ABC linear triblock copolymers, we studied in detail the effects of interaction parameters upon phases with fixed volume fractions of each segment. For simplicity, our focus are located on symmetric samples,χABN=χBCN,fA=fC. We have studied three samples, fA=fC=0.201, fA=fC=0.25 and fA=fC=0.18. For the first one, four different ordered phases have been obtained:alternating sphere, alternating cylinders, alternating gyroid, alternating diamond and double gyroid, while there are two phases including alternating gyroid and lamellae for the second sample. For the third, lamellae and a decorated phase, spheres on lamellae are computed with various interaction parameters. The physical mechanism underlying these phase transitions with varying interaction parameters is discussed.
As a conclusion, we proposed a generic Fourier-space method for solving SCFT equations. This method combines the capabilities of searching new structures in block copolymers and determination of space groups of ordered phases. With this method, equilibrium phase behaviors of ABC star-shaped triblock copolymers and ABC linear triblock copolymers have been investigated, where phase diagrams are obtained.
当前,自洽平均场理论已经成为研究高分子体系相平衡态结构问题十分重要的理论方法。该方法最初由英国理论物理学家Edwards于上世纪60年代中期提出,随后由Helfand, Noolandi等将该理论引入到了嵌段共聚物等多相高分子体系中。过去十几年来,对探索自洽场理论数值解法的努力大大推进了该理论在高分子科学领域中的应用。目前发展成熟的自洽场理论数值解法包括：Matsen-Schick谱方法、Fredrickson-Drolet实空间解法和Rasmussen等提出的准谱方法,它们分别提出于上世纪九十年代中期、末期和本世纪初。这些方法各有优缺点。Matsen-Schick谱方法中,空间变量函数以某些具有设定空间群对称性的基函数展开,进而在Fourier空间求解自洽场方程。该方法适用于计算所设定结构对称性的形态的自由能,在计算体系相图方面优势显著,但不适用于新形态的预测。Drolet-Fredrickson实空间方法和准谱方法等虽然可以预测嵌段共聚物的新结构,但是计算大尺寸体系时较难得到十分规整有序的结构。而且,对那些复杂双连续型网络结构(如Gyroid、金刚石结构、Fddd等)的搜索不仅十分困难,在确定这些复杂结构的对称性空间群方面更有所欠缺。最近,鉴于在线型嵌段共聚物中发现的有序结构多为中心对称结构,Guo等提出了一种可用于搜索中心对称结构的具有一定“普适性”的谱方法。在该方法中,所有空间变量函数以余弦函数展开,进而在Fourier空间进行求解自洽场方程。然而,在嵌段共聚物某些体系中,如星型共聚物体系、嵌段共聚物共混体系等,会有出现非中心对称结构。因此,对具有非中心对称结构的嵌段共聚物体系的新结构预测方面Guo等发展的方法将不再适用,需要继续发展新的方法。在此背景下,本论文主要目的是发展一种具有真正普适性的方法：不仅能搜索嵌段高分子体系所有可能出现的结构(包括中心对称结构和非中心对称结构),又能够从计算结果中分析所得结构的对称性方面的信息,从而方便地确定一些复杂结构的对称空间群。将这些结构作为备选态采用传统的谱方法进行自由能的精确计算,有望可以得到体系的精确相图。为此,我们进一步拓展了Guo等的谱方法。其基本思想是：基于嵌段共聚物可以形成规则的有序周期性结构,我们可以将与空间有关的所有变量进行Fourier级数(包括余弦函数和正弦函数)展开,使自洽场理论方程均在Fourier空间求解。该方法可以近似看作：起初我们给定一个具有所有空间群对称性的“基函数库”,随着整个计算迭代过程进行,会自动筛选出与设定体系分子参数相匹配的那些具有某些对称性的基函数。从而实现搜索新结构的功能。本方法的缺点是,由于空间变量的Fourier展开项个数受限于计算能力的制约,在计算过程中,只能取有限个基函数展开。这将会给计算结果的准确性带来影响。
方法建立后,我们对其可靠性和有效性进行了检验。首先,采用此方法考察了最简单的AB两嵌段高分子体系。我们成功得计算得到了AB嵌段共聚物本体体系中已知的四种结构：层状结构、六角柱状结构、体心立方球状结构和Gyroid双连续结构。然后,我们将该方法用于ABC三嵌段高分子的形态的预测。同样,我们成功的预测出了线型高分子体系中已经实验观察到和理论预测出来的一些典型结构,如交替型结构、核壳型结构、复合型结构等等。最后,对ABC星型高分子体系的结构预测做了考察。计算得到了该体系中典型的几种二维柱状多边形结构：[6.6.6],[8.8.4],[8.6.4；8.6.6],[10.6.4；10.6.6],[12.6.4；8.6.4],[12.6.4]等。综合对以上几个体系的检验,我们得出结论：本论文中发展的自洽场理论Fourier空间解法具备嵌段高分子体系中的结构预测功能,可以被用来研究和搜索嵌段共聚物中的新的有序相结构。
在嵌段共聚物相分离实验中,X射线衍射方法是测定其形成的有序结构的重要方法。根据散射理论,我们推导出了以Fourier空间数值解的具体形式表达的体系散射函数。以此为基础,分别讨论了AB两嵌段高分子、ABC线型和星型嵌段高分子体系中已知的有序结构的散射性质,得到了这些结构的散射强度与散射波矢模的关系图谱,将这些计算结果同实验结果进行了对比,在这些有序结构(包括已知的那些复杂性网络结构)的散射特征方面理论计算和实验结果取得了十分一致的结论。这些结果表明：该方法具备确定有序态结构的能力,包括对于那些复杂的双连续网络型结构的对称性的确定,从而使得该方法在预测复杂嵌段共聚物相形态方面具有各大的优势。
ABC星型嵌段共聚物由于其特殊的链拓扑结构,赋予了它们十分独特的自组装行为。鉴于目前对该体系的本体组装行为理解还不是十分深入,为了较好的认识该类型嵌段高分子的本体中的组装形态,我们采用前面提出的自洽场理论Fourier空间解法对ABC星型嵌段共聚物体系的本体平衡态相分离形态做了详细的考察。为了提高计算效率,并考虑到ABC星型高分子在三个支链链长相当时多形成二维柱状多边形结构,所以我们将自洽场计算限定在二维空间中。我们主要考察了两类星型共聚物体系。第一类为三组分具有相同的相互作用,我们称之为对称相互作用体系,χABN=χBCN=χACN=30.0,第二类则为非对称相互作用体系,χABN=χBCN=25.0,χACN=37.0。我们分别计算得到了这两类体系的三元相图。在对称相互作用体系中,计算得到了六种二维柱状多边形结构,包括：[6.6.6]、[8.6.4；8.6.6]、[8.8.4]、[10.6.4；10.6.6]、[12.6.4；8.6.4]和[12.6.4]；将该计算结果同Monte Carlo模拟、DPD模拟等方法的结果进行了比较,得到了一致的结果。在第二类体系中,则只得到了四种有序结构：[8.6.4；8.6.6]、[8.8.4]、[10.6.4；10.6.6]和[12.6.4]。我们将第二类体系计算结果同实验中ISP体系发现的结构做了对比,发现在相转变规律上得到了一致的定性结论,但在定量的相转变点上存在一些不一致性。这种不一致性可能是由于自洽场计算中所设定的分子参数等同实验实际体系不一致的原因。
相比于AB两嵌段共聚物体系来说,ABC三嵌段高分子体系具有更多可调参数,这就赋予了该体系更加丰富且更难穷尽的有序结构。在ABC线型嵌段高分子本体相行为方面,实验和理论都做了一些有意义的工作。实验中主要详细考察了两个体系：ISO和SBM (SEBM)体系,理论计算等方面目前已得到的两个相图也均是对应于这两类体系。这些实验和理论研究工作均是给出了固定链段间相互作用参数情况下,不同的组分比例而对应的平衡态相结构的信息。为了从另外一个侧面对ABC三嵌段高分子体系本体相行为作一些深入了解,即固定组分比例而改变链段间的相互作用参数,考察体系的有序结构将怎样变化,我们主要考察了对称体系中,即χABN=χBCN,fA=fC,相互作用参数χABN和χACN的变化对嵌段高分子的平衡态结构的影响。为了提高数值计算稳定性和计算效率,我们将体系限定在弱分凝至中等强度分凝情况下进行考察。我们探讨了三个不同组分体积分数时的情况,分别是：fA=fC=0.201、fA=fC=0.25和fA=fC=0.18。我们发现,在不同的组分相互作用情况下,第一个体系可以形成多种不同的有序结构,包括交替型球状相、交替型柱状相、交替型Gyroid、交替型金刚石结构和Gyroid结构。对于第二类体系,通过改变相互作用参数,我们得到了交替型Gyroid和三层层状结构两种形态。第三类体系计算结果表明,在不同的组分相互作用参数下,体系可以发生从层+球结构向三层层状相的转变。我们分别对这些相转变过程的物理机制做了初步的探讨。
综上所述,本论文发展了自洽场理论Fourier空间解法,本方法同时具备预测新结构和确定所得结构对称性信息的功能。应用该方法我们分别研究了ABC星型嵌段共聚物和ABC线型嵌段共聚物的本体相行为,计算得到了它们的相图。
Self-organization of soft condensed matters has been one of the hot topics in the field of condensed matter physics. Full understanding of their structural formations, along with their dynamics features during phase transitions will make contributions to design novel functional materials. Soft matters include polymers, colloids, surfactant, biomacromolecules etc. As a typical soft matter, block copolymers have taken considerable attentions for decades due to their fruitful nano-scale ordered structures and potential applications. It has been well known that there are five equilibrium ordered phases in block copolymer melts, alternating lamellae, hexagonally-packed cylinders, center-body cubic spheres, double gyroid and Fddd (O70).
Self-consistent field theory (SCFT) has been one of the most important and successful theories in polymer science. The SCFT has its origin in the work of Edwards in 1960s. Later, this theory was adapted explicitly to treat block copolymers by Helfand and Noolandi etc. During last decades, establishment of various numerical methods for solving SCFT equations, which include "spectral method" proposed by Matsen and Schick in 1994, "real-space method" proposed by Drolet and Fredrickson in 1999 and "pseudo-spectral method" proposed by Rasmussen etc. in 2001, have greatly improved the applications of SCFT in the field of polymer science. These methods have their advantages and disadvantages. Take "real-space method" for instance, although this method is good at searching new ordered structures in block copolymers, difficulties are encountered at reaching long-range ordered structures and determination of space group of complex ordered phases, especially those of complex networks, like gyroid and Fddd. For Matsen-Schick spectral method, all spatially varying functions are expanded in terms of basis functions with priori symmetry and SCFT equations are cast in Fourier space. Consequently, the spectral method is only good at computation of phase diagrams. Recently, Guo et al proposed a generic method for solving SCFT equations, in which all spatially varying functions are expanded in terms of cosine functions, provided that most of ordered phases observed experimentally are centrosymmetric. There are, however, some non-centrosymmetric structures in block copolymer melts, especially in star-shaped terpolymers, where the generic method for SCFT is no longer valid. Under this situation, we aim at proposing a generic approach to solution of self-consistent field theory (SCFT) equations for block copolymers, which combines the capabilities of searching new ordered phases (including centrosymmetric and non-centrosymmetric structures) and determination of space groups of obtained structures together. In this method, all spatially varying functions are expanded in terms of Fourier series (including cosine and sine functions) which are essential determined by computational box parameters. Then, SCFT equations can be cast in terms of expansion coefficients. This method can be looked as an expansion of Matsen-Schick spectral method. The advantage of the approach consists in fact that structural symmetries of resulting ordered phases can be easily deduced from expansion coefficients of nonzero values, which will be clearly demonstrated for complex phases in block copolymers.
To evaluate the capabilities of generic Fourier-space method for SCFT, we firstly applied this method to AB diblock copolymer melts. As expected, we successfully obtained the ordered phases including alternating lamellae, hexagonally-packed cylinders, body-centered cubic spheres and double gyroid. Then, the equilibrium ordered phases in ABC linear triblock copolymers melts are computed using the generic Fourier method proposed. Some typical phases, including alternating type, core-shell type and the decorated type structures, are obtained successfully. For some non-centrosymmetric structures in ABC star-shaped terpolymers, we extend the Fourier method to the equilibrium phases in ABC star-shaped terpolymers. Some two-dimensional cylindrical structures are computed, among which are [6.6.6], [8.8.4], [8.6.4;8.6.6], [10.6.4;10.6.6], [12.6.4;8.6.4], [12.6.4]. Note that [6.6.6] and [10.6.4; 10.6.6] are two typical structures of non-centrosymmetric space group. With the above verifications, a conclusion is reached that the generic Fourier method for SCFT is capable of searching new ordered structures with any space groups.
Structural determinations of ordered phases in experiments are made via small-angle X-ray scattering (SAXS) method. We have mentioned that the Fourier method proposed aims at combination of capabilities of searching new structures and of determination of space-groups of obtained ordered phases. In order to make a close comparison with SAXS results in experiments, derivations of the scattering intensities of ordered phases in terms of Fourier coefficients in our method is made. Then, the scattering intensities of some typical ordered morphologies in AB diblock copolymers and ABC triblock copolymers are computed and compared with those in experiments. A good consistence between the theoretical scattering functions and those in experiments proves the capability of determination of space groups of ordered structures in block copolymers.
ABC star-shaped terpolymers have attracted attentions for years due to their fruitful ordered morphologies. The most distinction of phase behaviors of star terpolymers from that of linear triblock copolymers lies in fact that junction points in ABC star terpolymers be arranged along one-dimensional lines resulting from topological constraint, while in linear ABC triblock copolymers two-dimensional plane could be allowed for the connecting points between neighboring blocks. The spatial arrangement of junction point in star-shaped terpolymers leads to two-dimensional cylinder-type morphologies under assumption that three polymer chains are totally incompatible and long enough. With the generic Fourier method for SCFT, the equilibrium phases of ABC star-shaped terpolymers have been studies. Two broad types are investigated in detail:one with symmetric interactions,χABN=χBCN=χACN=30.0, and one with asymmetric interactions,χABN=χBCN=25.0,χACN=37.0, corresponding to ISP star-shaped terpolymers. The triangle phase diagrams are obtained. For the former, six ordered morphologies are obtained, including [6.6.6], [8.6.4;8.6.6], [8.8.4], [10.6.4; 10.6.6], [12.6.4;8.6.4] and [12.6.4], which is in consistence with those simulations by Monte Carlo and DPD. For asymmetric interactions, two series of star terpolymers are studied, A1.0B1.0Cx, A1.0B1.8Cx. After comparison with experimental results for ISP, a qualitative agreement is reached, while there are few quantitative agreements. These discrepancies maybe due to the inconsistence of molecular parameters used in SCFT with those in real star terpolymers, which are hardly determined experimentally for now, like Flory-Huggins interaction parameter, statistical segment length, etc.
ABC linear triblock copolymers reveal fruitful phase behaviors in comparison with AB diblock copolymers, due to their vast parameter space. Thanks to efforts of experimental and theoretical workers on block copolymers, some knowledge has been collected on equilibrium phase behaviors of ABC linear triblock copolymers. Only few samples of ABC linear triblock copolymers, however, have been studied in detail, including ISO and SBM (SEBM). The phase diagrams that have been computed theoretically are for these two types of triblock copolymers, which provide the information of equilibrium phases under each molecular parameter. For obtaining another insight into phase behaviors of ABC linear triblock copolymers, we studied in detail the effects of interaction parameters upon phases with fixed volume fractions of each segment. For simplicity, our focus are located on symmetric samples,χABN=χBCN,fA=fC. We have studied three samples, fA=fC=0.201, fA=fC=0.25 and fA=fC=0.18. For the first one, four different ordered phases have been obtained:alternating sphere, alternating cylinders, alternating gyroid, alternating diamond and double gyroid, while there are two phases including alternating gyroid and lamellae for the second sample. For the third, lamellae and a decorated phase, spheres on lamellae are computed with various interaction parameters. The physical mechanism underlying these phase transitions with varying interaction parameters is discussed.
As a conclusion, we proposed a generic Fourier-space method for solving SCFT equations. This method combines the capabilities of searching new structures in block copolymers and determination of space groups of ordered phases. With this method, equilibrium phase behaviors of ABC star-shaped triblock copolymers and ABC linear triblock copolymers have been investigated, where phase diagrams are obtained.
引文
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[10]Hajduk, D. A.; Harper, P. E.; Gruner, S. M.; Honeker, C. C.; Kim, G.; Thomas, E. L.; Fetters, L. J. The Gyroid-a New Equilibrium Morphology in Weakly Segregated Diblock Copolymers[J]. Macromolecules 1994,27, (15),4063-4075.
[11]Forster, S.; Khandpur, A. K.; Zhao, J.; Bates, F. S.; Hamley, I. W.; Ryan, A. J.; Bras, W. Complex Phase-Behavior of Polyisoprene-Polystyrene Diblock Copolymers near the Order-Disorder Transition [J]. Macromolecules 1994,27, (23),6922-6935.
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