JOURNAL OF MATHEMATICAL PHYSICS 46, 013301 (2005)
Bifurcation analysis for the construction of a phase diagram of heteropolymer liquids A. N. Ivanova Institute of the Problems of Chemical Physics, RAS, Moscow, Russia
S. I. Kuchanov Institute of Applied Mathematics, RAS, Moscow, Russia
L. I. Manevitcha) Institute of Chemical Physics, RAS, Moscow, Russia (Received 10 June 2003; accepted 23 August 2004; published online 27 December 2004)
Rather general mean field theory of heteropolymer liquids developed earlier reduces the problem of the phase diagram construction to the determination of extremals of the free energy functional. These should be subsequently analyzed for their local and global stability. Tackling of this problem traditionally involves the examination of the behavior of the solutions of a set of nonlinear algebraic and partial differential equations at various values of the control parameters. Besides, the necessity arises here to construct in space of these parameters the lines where a polymer system loses the thermodynamic stability. To overcome mathematical difficulties encountered we employed a complex approach that combines analytical and numerical methods. A two-step procedure constitutes the essence of such an approach. First, the bifurcation analysis is invoked to find the asymptotics of the extremals in the vicinity of bifurcation points. Then these asymptotics are used as an initial approximation for the numerical continuation of specific lines, where the stability loss occurs, into regions of the parametric space far removed from bifurcation values. We realized this approach for the melt of linear binary copolymers of various chemical structure with macromolecules having a pattern of arrangement of monomeric units describable by a Markov chain. Bifurcation and phase diagrams for some of these copolymers have been constructed within a wide range of temperatures and volume fractions of a polymer. © 2005 American Institute of Physics. [DOI: 10.1063/1.1827323]
I. INTRODUCTION
The theoretical physics of polymers in its current state suggests the application of a rather sophisticated mathematical method (see, for instance, Ref. 1). This is because the majority of differential and integral equations which describe polymer systems are nonlinear admitting therefore several physically meaningful solutions.2 Consequently, problems of nonlinear analysis of these equations based on the approaches of the theory of bifurcations, typical for the mathematical physics, are usually encountered here. One of such nontrivial problems particularly important for the thermodynamics of polymers is attacked in the present paper. The methods of its solution may be of interest for the physicist–theorists dealing with the Landau theory of phase transitions and the statistical physics of disordered systems. The experts in the field of the applied mathematics may also benefit from getting familiar with the solution of this problem, that may prompt them to look for new possible applications of the contemporary methods of the nonlinear analysis in the theoretical physics of polymers.
a)
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013301-2
J. Math. Phys. 46, 013301 (2005)
Ivanova, Kuchanov, and Manevitch
The calculation of a phase diagram belongs to the most challenging theoretical problems of the thermodynamics of melts and solutions of polymers.3 Having constructed such a diagram one might judge about the phase state of a polymer liquid under thermodynamic equilibrium at given values of external parameters. A crucial feature of polymer systems stipulating their qualitative distinction from low-molecular ones is the possibility of the existence of mesophases. Each of them represents such an equilibrium state of the liquid of macromolecules comprising more than one type of elementary units where the density of these latter periodically changes in space at scales lying between atomic and macroscopic length scales. Depending on this density profile dimensionality d a mesophase spatially periodic structures can vary in symmetry. The most investigated among them are lamellar 共d = 1兲, hexagonal 共d = 2兲, and body centered cubic 共d = 3兲 structures. They have been found experimentally and scrutinized theoretically.4 The type of a heteropolymer equilibrium structure as well as its period and amplitude are controlled along with temperature and pressure also by architecture, composition, and structure of macromolecules. Most theoretical and experimental research addressed monodisperse block copolymers, in which all molecules being identical consist of two or three sufficiently long blocks of elementary units.5 However, synthetic polymers represent as a rule a mixture of macromolecules markedly distinguishing in the content of various units and in the pattern of their arrangement along polymer chains. Thus the number of types of macromolecules in a real polymer is virtually infinite for any polymer specimen. That is why the description of its chemical structure suggests the recourse to some statistic approach. By the most general of them the set of macromolecules constituting a linear copolymer specimen is presumed to be mapped onto the set of realizations of a stochastic process.6 It implies the transition from a particular monomeric unit of a macromolecule to the next one at every unit interval of “time.” The role of the regular state S␣ 共␣ = 1 , . . . , m兲 is played here by ␣th type unit while the transition into absorbing state S0 corresponds to going out of a macromolecule. Such a stochastic process with discrete time and finite number of states is referred to as a stochastic chain. The best known among them is the Markov chain where the probability to fall into any state at a certain step is exclusively controlled by the type of the state at the preceding step.7 This absorbing chain is characterized by the matrix of transition probabilities Qab =
冋 册 1
0
0 Q
.
The element ␣ of matrix Q equals the probability of the transition from regular state S␣ into regular state S whereas row vector 0 and column vector 0 have components 0␣ = 0 and ␣0, respectively. Probability of the absorption ␣0 can be expressed through elements of matrix Q from the normalization condition ␣0 = 1 − 共␣1 + ¯ + ␣m兲. Hence matrix Q and vector of initial states v with components v␣ completely specifies a Markov chain. Nowadays it is established that the chemical structure of many synthetic copolymers is described by a Markov chain. This stipulates practical importance of the investigation of their thermodynamic behavior. Besides, relationships have been derived that express the matrix of transitions Q and vector v through kinetic parameters of a reaction system where copolymers are synthesized.6,8 Because these parameters are reported for a great number of particular copolymerization processes,8 an opportunity opens up to calculate phase diagrams of real copolymers formed in the course of these processes. In the last decade a number of theoretical works have been published devoted to the description of spatially periodic structures formed in melts of binary Markovian copolymers.9–16 The approach employed in these papers is a variation of the Landau theory of the phase transitions. This is based on the expansion of a system free energy in powers of the order parameter and on cutting off all terms whose power is more than four. Evidently, such a procedure is correct only in the vicinity of the critical point where the order parameter is sufficiently small. Thus the region of applicability of the phase diagrams presented earlier9–16 is restricted just to this narrow range of the copolymer melt parameters. To have phase diagrams constructed within the whole range of external parameters an original approach was put forward which relies on the description of the nuclei of the incipient phase.17 This approach enabling one to relax the Landau theory restrictions suggests finding nontrivial
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013301-3
J. Math. Phys. 46, 013301 (2005)
Bifurcation analysis for the construction
solutions of a set of either integral or differential equations with the subsequent analysis of their stability. A major task of this paper is to elucidate the potentialities of the practical implementation of this approach for the construction of the phase diagram of the Markovian copolymer melt using mathematical apparatus of the bifurcation theory. Bifurcation analysis was earlier used for the construction of nonhomogeneous unstable structures occurring in the polymer blends.18,19 The efficiency of the bifurcation analysis as applied to the theory of polymer liquids has been demonstrated earlier when describing the dynamics of the phase transitions in a mixture of two homopolymers,18,19 each consisting of the identical molecules. This system essentially differs from that addressed in the present paper. Qualitative distinction between these two systems lays in the fact that the first of them is a two-component one unlike the second system that comprises virtually infinite number of components. The last circumstance appreciably complicates the thermodynamic description of heteropolymer liquids considered in the present work as compared to the homopolymer ones. II. PHASE DIAGRAM
We consider the melt of a binary copolymer whose macromolecules have chemical structure described by a Markov chain. This is completely specified by five independent parameters 11, 12, 21, 22, and v1 or v2, through which any statistical characteristic of a copolymer primary structure is expressed. In particular, fractions X01 and X02 of the first and the second type units as well as their average number in a macromolecule, PN0 , can be calculated by formulas X01 =
⌬1 , ⌬
X02 =
⌬2 ; ⌬
⌬ = ⌬ 1 + ⌬ 2,
Y0 ⬅
1 PN0
=
D , ⌬
共1兲
where the following designations are used: ⌬1 = 21 + v120,
⌬2 = 12 + v210,
D = 1220 + 2110 + 1020 .
共2兲
Along with “chemical” parameters ␣ and v␣ a copolymer melt is characterized by a set of “physical” parameters. Among the latter in the framework of the model in hand are temperature T, pressure P, and parameters ␥␣ 共␣ ,  = 1 , 2兲 of pair interaction between ␣th and th type units. With bifurcation analysis in mind it is more convenient to use parameter ⌽0, equal to the volume fraction occupied by monomeric units in principal phase, rather than the pressure. This parameter is simply expressed ⌽0 = M 0v / V through volume v of a monomeric unit, their number M 0 in a system and its volume V. Quantity ⌽0 occurring in most theories of polymer liquid3,20,21 is uniquely related to P by the equation of state. Key elements of the phase diagram of such a liquid are two curves, each representing a two-dimensional section of the hypersurface (whose codimension is 1) in the space of external parameters. The first of them, cloud point curve (CPC), is a locus of points where nuclei of incipient phase or mesophase become globally thermodynamically stable. According to which of these cases takes place the CPC branch is said to be trivial or nontrivial. At the second of the above-mentioned curves, termed spinodal, the spatially homogeneous state loses local thermodynamic stability. The spinodal curve (SC) can contain trivial and nontrivial branches depending on the spatial scale (wave vector qជ = ជq*) of the density fluctuations at which the loss of the above ជ * = 0兲 while to nonstability happens. To a trivial branch there corresponds macroscopic scale 共q trivial one mesoscopic scale 共qជ * ⫽ 0兲 conforms. Provided they coexist, these two branches are tangent at the Lifshitz point. A domain of a phase diagram located in between CPC and SC corresponds to the metastable state of the principal phase. The density of each type of units in the incipient phase nuclei has at critical point (where CPC and SC have common tangent) the same value as in the principal phase. In order to construct the phase diagram of a copolymer melt we will proceed from the general thermodynamic theory of heteropolymer liquids.17,22 According to this theory the thermodynamic description of a polymer liquid is performed in terms of density functional
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013301-4
J. Math. Phys. 46, 013301 (2005)
Ivanova, Kuchanov, and Manevitch
⌬F关⌽兴 ⬅ F关⌽兴 − F关⌽0兴 = TM 0兵G关1兴 − G关s兴其 +
冕
drជ兵f * 共⌽兲 − f * 共⌽0兲其 +
1 v
冕
drជ兵H0⌽0 − H⌽其. 共3兲
Each ␣th component of vectors ⌽ , H, and s, entering into the right-hand part of this expression refers to the ␣th type monomeric unit. So ⌽␣ denotes the volume fraction of these units in nuclei while H␣ is the component of external field H acting upon them. Superscript “0” means that the values of these quantities are taken in the principal phase. As for the argument of generating functional G关s兴 it has components s␣ = exp兵共H␣0 − H␣兲 / T其. For the case of Markovian copolymers this functional can be presented as follows:
G关s兴 =
1 V
冕
drជ共s1U110 + s2U220兲 =
Y0 V
冕
drជ共s1V1v1 + s2V2v2兲,
共4兲
where dependence of U and V on s is found from the set of four equations, KU␣ − 共U1s11␣ + U2s22␣兲 = v␣Y 0
共␣ = 1,2兲,
KV␣ − 共␣1s1V1 + ␣2s2V2兲 = ␣0 .
共5兲
Linear operator K in the left-hand side of Eqs. (5) is inverse to the integral operator with kernel 共rជ − rជ⬘兲 which has the sense of the conditional probability for the neighbor of unit situated at point rជ⬘ to be at point rជ. Kernel is a rapidly decreasing function which vanishes at distance 兩rជ − rជ⬘兩 comparable with a monomeric unit size a. Since essentially larger spatial scales are of actual practical interest, any normalized function satisfying the above condition might be chosen ជ兩. as . Its Fourier transform x ⬅ ˜共q兲 is governed exclusively by the wave vector modulus q ⬅ 兩q Being positive quantity x turns into unity at q = 0 and vanishes at q → ⬁. Function ˜共q兲 = exp共−a2q2 / 6兲, traditionally employed for numerical calculations, meets these conditions. Functional (3) is controlled by dimensionless units’ densities 兵⌽␣其 both explicitly and implicitly through field H. Its dependence on ⌽ is determined from expressions ⌽ 1 = ⌽ 0s 1U 1V 1,
⌽ 2 = ⌽ 0s 2U 2V 2 ,
共6兲
which are obtained from the conditions of vanishing of the first order variational derivatives of functional (3) with respect to H1 and H2. As for the explicit dependence of ⌬F (3) on ⌽, this is specified by function f * 共⌽兲 having the following appearance:17,21 f * 共⌽兲v 1 = 共1 − ⌽兲ln共1 − ⌽兲 + ⌽ − 共␥11⌽21 + ␥22⌽22 + 2␥12⌽1⌽2兲, T T
共7兲
where ⌽ = ⌽1 + ⌽2. Hence, expressions (4) and (7) along with equations (5) and (6) completely define density functional (3) defined on the set of smooth functions ⌽1共rជ兲 , ⌽2共rជ兲 lying within region 0 艋 ⌽1 , ⌽2 ⬍ 1 , ⌽1 + ⌽2 ⬍ 1. Equilibrium spatial distributions ⌽e1共rជ兲 and ⌽e2共rជ兲 of the units’ volume fractions in nuclei are found by the minimization of the density functional. Necessary conditions of its minimum are v␦F关⌽兴
␦⌽␣共rជ兲
= − H␣共rជ兲 + ␣* 共⌽共rជ兲兲 = 0
共␣ = 1,2兲,
共8兲
where the following designation is used:
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013301-5
J. Math. Phys. 46, 013301 (2005)
Bifurcation analysis for the construction
␣* 共⌽兲 ⬅
v f * = − T ln共1 − ⌽兲 − 2共␥␣1⌽1 + ␥␣2⌽2兲. ⌽␣
共9兲
Expression (8) establishes the relation between H共rជ兲 and ⌽共rជ兲 taken on functional ⌬F extremal. With this relation in mind one can find dimensionless equilibrium densities of units in nuclei from the solution ⌽1 = ⌽e1 , ⌽2 = ⌽e2 of equations (5) and (6), where vector s components read as s␣ = exp兵关␣* 共⌽0兲 − ␣* 共⌽兲兴/T其.
共10兲
These equations at any values of external parameters have the trivial solution U␣共rជ兲 ⬅ U␣0 = X␣0 ,
V␣共rជ兲 ⬅ V␣0 = 1,
⌽␣共rជ兲 ⬅ ⌽␣0 = X␣0 ⌽0,
s␣共rជ兲 ⬅ 1
共11兲
describing spatially homogeneous state of a system. It becomes thermodynamically absolutely unstable on the spinodal, whose mathematical condition is the loss of the positive definiteness by matrix M0共x兲 = 关⌽0X0共x兲兴−1 − C0v−1 ,
共12兲
where x is a dummy variable laying within the interval between 0 and 1. The right-hand side of this expression comprises copolymer structure matrix X0共x兲 and matrix of direct correlation functions C0 ⬅ C共⌽0兲 describing the melt of monomeric units. In the framework of the simplest “lattice liquid” model21 elements of the latter matrix look as 2␥␣ 1 C␣共⌽兲 v2 f * 共⌽兲 + . ⬅− =− T ⌽ ␣ ⌽  1−⌽ T v
共13兲
As for the structure matrix its elements in a simple way17 0 0 0 X␣ 共x兲 = X␣0 ␦␣ + W␣ 共x兲 + W␣ 共x兲
共14兲
are related to the elements of matrix W0共x兲 of generating functions of two-point chemical correlators of macromolecules in the principal monophase system. For a Markovian copolymer the following expressions hold: 0 0 共x兲 = X␣0 L␣ 共x兲, W␣
where L0共x兲 = 共E − xQ兲−1xQ,
共15兲
which in combination with expressions (13) and (14) completely define matrix (12). This loses the positive definiteness when its minimal eigenvalue vanishes. If it happens at point x = 1 or at any other point within interval 0 ⬍ x ⬍ 1 we are dealing with trivial or nontrivial branch of SC, respectively. The explicit equations enabling one to find both branches of the spinodal of a binary Markovian copolymer in terms of the elements of matrices X0共x兲 and C0 have been presented and analyzed earlier.9,10 At some values of the input parameters, the set of equations (5), (6), and (10) can have solutions differing from the trivial one (11). Among these nontrivial solutions ⌽ = ⌽e corresponding to the density functional extremals only those have physical meaning on which this functional has a minimum. Just such solutions describe the nuclei of the incipient phase. A necessary and sufficient condition for a spatially homogeneous nontrivial solution of Eqs. (5), (6), and (10) to provide a minimum of the density functional is the positive definiteness of matrix Me共x兲 = 关⌽0Xe共x兲兴−1 − Cev−1 ,
共16兲
where Ce ⬅ C共⌽e兲 while elements of matrix Xe共x兲 are defined by the following expression: e e e 共x兲 = 关⌽␣e /⌽0兴␦␣ + W␣ 共x兲 + W␣ 共x兲 X␣
共17兲
and the designation is employed
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013301-6
J. Math. Phys. 46, 013301 (2005)
Ivanova, Kuchanov, and Manevitch e e W␣ 共x兲 = s␣e se U␣e L␣ 共x兲Ve .
共18兲
Here s␣e is found by formulas (9) and (10) via substitution of quantities ⌽␣e for ⌽␣ whereas given ⌽␣e values of U␣e and V␣e represent the solution of a set of four linear algebraic equations (5). As e for matrix Le共x兲 whose elements are L␣ 共x兲 it has the following appearance: Le共x兲 = 共E − xQS兲−1xQ,
共19兲
where S is the diagonal matrix with elements s␣␦␣. Because Eqs. (5), (6), and (10) are nonlinear they can have more than one solution ⌽e at the same values of external parameters. Among these extremals only those have physical meaning which provide positive definiteness of matrix (16). Each such solution, ⌽i, specifies volume fractions ⌽i1 and ⌽i2 of monomeric units of the ith phase nuclei. Equations (5), (6), and (10) can have solutions ⌽e共rជ兲 taking on different values at different points rជ of the Euclidean space. Among these extremals those should be chosen which minimize the density functional. To make this choice it is convenient to consider its second variation represented as a quadratic functional with respect to variations of volume fractions of monomeric units. Such a treatment will be realized in the next section. Having found all local minima of the density functional it is necessary to reveal among them the global one. Equating the value of the above functional in this minimum to its value in the minimum at the trivial solution we will get the condition for finding the cloud point hypersurface. This may be formulated mathematically in terms of the functional R关⌽i兴 =
冕
再
2
drជ T⌽0Y 0
兺 V␣共1 − s␣i ␣i 兲 + v关P * 共⌽0兲 − P * 共⌽i兲兴 ␣=1
冎
,
共20兲
where the following designation is used: v P * 共⌽兲 = − T关ln共1 − ⌽兲 + ⌽兴 − 共␥11⌽21 + ␥22⌽22 + 2␥12⌽1⌽2兲.
共21兲
Conditions for determining CPC are evident, R关⌽I兴 = 0,
R关⌽i兴 ⬎ 0
at i ⫽ I,
共22兲
where superscript I specifies the global minimum of the density functional. Depending on whether ⌽I is homogeneous or spatially periodical solution of Eqs. (5), (6), and (10) the conditions (22) refer to the trivial or nontrivial CPC branch, respectively. For the first of them, functional R关⌽I兴 reduces to function R共⌽I兲 vanishing when the pressure in nuclei and the principle phase is the same. All mathematical formulas presented in the foregoing are valid for copolymers whose macromolecules are described by an arbitrary Markov chain. When it is symmetric these formulas become noticeably simpler. The condition of such a symmetry is v11220 = v22110 .
共23兲
For statistically symmetric Markovian copolymers the following expressions hold: v1 =
X01 =
2110 , 2110 + 1220
21 , 12 + 21
X02 =
v2 =
12 , 12 + 21
1220 , 2110 + 1220 Y 0 = X0110 + X0220 .
共24兲
共25兲
Interestingly, the composition vector X0 of such copolymers coincides with the stationary vector of the nonabsorbing ergodic Markov chain7 describing macromolecules of infinite length. From
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013301-7
J. Math. Phys. 46, 013301 (2005)
Bifurcation analysis for the construction
the symmetry condition (23) there follows the proportionality of the components of vectors U and V: U1共rជ兲 = 1V1共rជ兲,
U2共rជ兲 = 2V2共rជ兲.
共26兲
Consequently, in case of symmetric Markovian copolymers the first pair of equations (5) may be omitted when finding the density functional extremals. Besides, the matrix of the generating functions of the two-point chemical correlators of such copolymers is symmetric so that equalities 0 0 e e 共x兲 = W␣ 共x兲 and W␣ 共x兲 = W␣ 共x兲 may be taken into account when considering relationships W␣ (14) and (17), respectively. Once the mathematical problem of a phase diagram construction is completely formulated let us address the bifurcation analysis of the above nonlinear equations. Of prime interest here is the behavior of their solutions at scales essentially larger than monomeric unit size a, which is the scale where function vanishes. With this in mind the integral operator in Eqs. (5) may be replaced by the differential one as it is customary in statistical physics of polymers.23 Thus, the first term in the second pair of equations (5) will read KV␣ ⬅
冕
共−1兲共rជ − rជ⬘兲V␣共rជ⬘兲drជ⬘ → V␣共rជ兲 −
a2 ⌬V␣共rជ兲, 6
共27兲
where ⌬ is the Laplace operator. Below, for convenience sake, instead of rជ we will use dimensionless variable 冑6rជ / a. III. THE BIFURCATION ANALYSIS
Construction of phase diagrams for heteropolymer liquids is a rather complicated global nonlinear problem. It is a reason why only Landau theory of phase transitions has been still applied to this problem, so only the vicinity of critical point could be correctly treated. In order to go beyond the Landau theory we have replaced the solution of global nonlinear problem by subsequent solution of local problems using nonlinear bifurcation analysis and continuation procedure by parameters fixing the temperature and characteristic size of the nonhomogeneous structure. We start with a determination of nontrivial solutions of the equations (5), (6), and (10) branching from trivial solution in vicinity of the spinodal points. For that we consider the nonlinear operator A共U , V , s , ⌽ , h兲, which is defined by the left-hand sides of equations (5), (6), and (10) depending on vector functions U, V, s, ⌽ as well as parameter h = 2␥12 / T. Let the vectors z = 关U共h兲 , V共h兲 , s共h兲 , ⌽共h兲兴 and z0 = 关U0共h兲 , V0共h兲 , s0共h兲 , ⌽0共h兲兴 correspond to the spatially homogeneous solution of these equations and h* to be a critical value of parameter h, corresponding to spinodal point. Linear operator L关z0 , h兴 being a differential of operator A with eigenvalue vanishing at h = h*, may be determined by the following relations [where vectors ␦z = 共␦U , ␦V , ␦s , ␦⌽兲, ␣ ,  = 1 , 2,]:
L共z ,h兲␦z: 0
冤
ⵜ 2␦ U ␣ − ␦ U ␣ +
兺 Q␣共s␦U + U␦s兲
ⵜ 2␦ V ␣ − ␦ V ␣ +
兺 Q␣共s␦V + V␦s兲
s␣−1␦s␣
−
兺 共− 1/共1 − ⌽兲 + h␥␣/␥12兲␦⌽
␦ ⌽ ␣ − ⌽ 共 ␦ s ␣U ␣V ␣ + s ␣V ␣␦ U ␣ + s ␣U ␣␦ V ␣兲 0
冥
.
共28兲
The components of the vector z0 determines in particular the trivial solution with relations (11). The systems of equations (5), (6), and (10) may be represented in the vicinity of z0 in the following form:
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013301-8
J. Math. Phys. 46, 013301 (2005)
Ivanova, Kuchanov, and Manevitch
A共z,h兲 = L共z0,h兲共z − z0兲 + F共z0,z − z0,h兲,
共29兲
where norm F 共z0 , z − z0 , h兲 is O共共norm共z − z0兲兲2兲. One can introduce matrix J共q兲,
J共q兲 =
冤
− 共q2 + 1兲E + Q * s
0
Q*U
0
0
− 共q + 1兲E + Qs
QV
0
0
0
−1
−C
− ⌽0sV
− ⌽0sU
− ⌽0UV
E
2
s
冥
共30兲
containing the two-dimensional blocks, sU, sV, UV being the products of diagonal matrixes with elements s␣, U␣, V␣. The elements of matrix C are determined by relations (21), E is unit matrix. The trivial spinodal h共⌽0兲 is determined by the equation det J共0兲 = 0, nontrivial one by two equations, det J共q兲 = 0, 共det J共q兲 / q2兲 = 0, and condition 2 det J共q兲 / 共q2兲2 ⬎ 0. In the case of trivial spinodal the eigenfunction g共r兲 of the operator L corresponding to vanishing eigenvalue is the eigenvector g1 of matrix J共0兲. In the case of nontrivial spinodal similar eigenfunction is the product of eigenvector g1 of matrix J共q兲 (with eigenvalue which is equal to zero) and eigenfunction of the Laplace operator under periodic boundary conditions with eigenvalue −q2. This eigenvalue in general is degenerate, i.e., the dimension of proper subspace is greater than 1 and depends on the symmetry of the considered domain. We will consider below the one-dimensional case and denote the first component of vector r by x. In this case the dimension of proper subspace is 2 and g共x兲 = g1共a cos共qx兲 + b sin共qx兲兲,
共31兲
where a and b are the arbitrary constants. In this case the degeneracy is the consequence of invariance of the periodic solutions with respect to translation. Later on for the component U1 we change the periodic conditions from U1 / x = 0 to x = 0 and x = 2 / q. Then it is possible to set b = 0. Dealing with bifurcation analysis we introduce the auxiliary parameter and represent the solution z = 共U , V , s , ⌽兲 of the equation 共32兲
A共U,V,s,⌽,h兲 = 0
and parameter h in vicinity of its critical value h* as asymptotic power series by subsidiary parameter , which characterizes the amplitude deviation as the unknown solution from the initial h = h * + · h共1兲 + 2h共2兲 + 3h共3兲 + ¯ , z = z共0兲 + z共1兲 + 2z共2兲 + 3z共3兲 + ¯ .
共33兲
We present the left-hand side of Eq. (32) as the Taylor series in the vicinity of value h = h* and vector z共0兲, z共0兲 = U0共h * 兲,
V0共h * 兲,
s0共h * 兲,
⌽0共h * 兲.
Then we select the terms of identical order with respect to and equate them to zero. The equations for z共1兲 can be written as follows: L共z共0兲,h * 兲z共1兲 = 0. 共1兲
共34兲
共0兲
Consequently, z is eigenfunction of operator L 共z , h * 兲 corresponding to zero eigenvalue. The following equations can be found recursively: L共z共0兲,h * 兲z共i兲 = i共z共1兲, . . . ,z共i−1兲,h共i−1兲兲.
共35兲 共i−1兲
The vector function i on the right-hand side of Eq. (35) is the linear function by h and the degree i polynomial of the components of vectors z共k兲 共k ⬍ i兲. The function i for i = 2 , 3, can be presented as
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013301-9
J. Math. Phys. 46, 013301 (2005)
Bifurcation analysis for the construction
2共z共1兲, . . . ,z共i−1兲,h共i−1兲兲 = − 1/2 兺 k,j
冉
3共z共1兲, . . . ,z共i−1兲,h共i−1兲兲 = − 1/2 兺 + 1/6
k,j
2F 共1兲 共1兲 2F 共1兲 共1兲 zj h , zk z j − 兺 z k z j j z jh
共36兲
2F 共1兲 共2兲 共2兲 共1兲 2F 共2兲 共1兲 共1兲 共2兲 共zk z j + zk z j 兲 + 兺 共z j h + z j h 兲 z k z j j hz j 3F
共1兲 共1兲 z共1兲 兺 j zk zl j,k,l z jzkzl
冊
共37兲
.
The operator L is Fredholm operator of index 0. Therefore the condition of the solution existence for Eq. (35) is the orthogonality of the vector function i to the eigenfunction corresponding to zero eigenvalue of operator conjugate to L 共z共0兲 , h * 兲 This condition determines the value of h共i−1兲. Such a procedure corresponds to Lyapunov–Shmidt method.24 In the case of multiple zero eigenvalues, z共1兲 is linear combination of linear independent eigenfunctions of operator L. Therefore it is necessary to require the orthogonality i to all independent eigenfunctions corresponding to vanishing eigenvalue of operator conjugate to L 共z共0兲 , h * 兲. These conditions determine h共i−1兲 and the coefficients of the linear combination of independent eigenfunctions. IV. BIFURCATION OF NONHOMOGENEOUS STRUCTURE FROM HOMOGENEOUS SOLUTIONS
Let det J共q * 兲 = 0 for the function z共0兲, which is the trivial or nontrivial homogeneous solution of the equation (32). We consider further the one-dimensional symmetrical case for m = 2. This case corresponds to the relations (23) and (24) and it is possible to consider the equation for V only. Then z共i兲 = 共V , s , ⌽兲 and in the first approximation z共1兲 = g cos共q * x兲, where g is the eigenvector corresponding to the zero eigenvalue of the matrix J共q * 兲. The function 2 can be written as follows:
冤
Qs共1兲V共1兲
2 = − 1/2共s共0兲兲−2共s共1兲兲2 − h共1兲␥⌽共1兲 +
冉
共1兲 ⌽共1兲 1 + ⌽2 共0兲 1−⌽
− 2⌽共0兲s共1兲V共0兲V共1兲⌸ − ⌽共0兲s共0兲共V共1兲兲2
冊
2
冥
,
where the following designations are used: ⌸ = diag关1,2兴,
共1兲 s共1兲 = diag关s共1兲 1 ,s2 兴,
共1兲 V = diag关V共1兲 1 ,V2 兴,
共2兲 ⌽共1兲 = diag关⌽共1兲 1 ,⌽2 兴.
The orthogonality condition leads to the relation h共1兲 = 0. The vector z共2兲 in the second approximation has the form z共2兲 = a + b cos共2q * x兲 + ahh共2兲 .
共38兲
The vectors a, b, ah are determined from the following equations: J共0兲a = r0,
J共2q * 兲b = r0,
J共0兲ah = rh .
共39兲
The expressions for vectors r0 and rh are presented in Appendix A. The vector rh is zero for bifurcation from trivial solution, therefore ah also is zero. The value h共2兲 can be determined from the orthogonality condition 3 with respect to the eigenfunction corresponding to the zero eigenvalue of the operator conjugate to L 共z共0兲 , h * 兲. Its expression is also presented in Appendix A. The type of bifurcation is determined by sign h共2兲. If h共2兲 ⬎ 0, then the bifurcation is supercritical and pair of nontrivial solutions exists for h ⬎ h*. For h共2兲 ⬍ 0 the bifurcation is subcritical, so that pair of nontrivial solutions exists for h ⬍ h*. When dealing with supercritical bifurcation in the point of nontrivial spinodal, nontrivial solution is stable; in the case of subcritical bifurcation
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013301-10
J. Math. Phys. 46, 013301 (2005)
Ivanova, Kuchanov, and Manevitch
it is unstable. The disposition of the trivial 共hst兲 and nontrivial spinodals 共hsn兲 is determined by the type of bifurcation in the point of trivial spinodal. If hst ⬍ hsn, a transcritical nontrivial homogeneous solution branches from trivial spinodal. If hst ⬎ hsn, the equation J共q兲 = 0 has two roots ql and qr 共ql ⬍ qr兲, between the points h = hsn and h = hst and, respectively, two pairs of branches of nongomogeneous extremals appear. At h = hst is ql = 0, and both nontrivial nonhomogeneous and nontrivial homogeneous solutions can branch from trivial spinodal. This case will be considered below. V. BIFURCATION IN THE SPINODAL POINT
Let hsn ⬍ hst, then zero is a multiple eigenvalue of the operator L 共z共0兲 , h * 兲. Simultaneously, det J共0兲 and det J 共q*兲 are equal to zero and g共0兲, g共q*兲 are eigenvectors of matrixes J 共0兲 and J 共q*兲 corresponding to zero eigenvalues. The first approximation can be expressed as follows: z共1兲 = ␣g共0兲 + g共q * 兲cos共q * x兲. The constants ␣ and  are found from the orthogonality condition of the right-hand side of 2 in (35) with respect to two eigenfunctions g共0兲T, gT共q * 兲cos共q * x兲 of operator conjugate to L 共z共0兲 , q * 兲. The equations for determination of the ratio ␣ /  and h共1兲 are written as
␣2共g共0兲T,r00兲 + 2共g共0兲T,r22兲 = − ␣h共1兲zh0 , ␣共gT共q * 兲,r02兲 = − h共1兲zh2 . Here 共f共x兲 , g共x兲兲 is the scalar product vector function in L2 关0 , l兴, the formulas for vectors r00, r02, r22 are presented in Appendix A, ˜ T兲, zh0 = 共␥˜g,g
˜ T共q * 兲兲, zh2 = 共␥˜g共q * 兲,g
where ˜g = 关g共0兲关5兴,g共0兲关6兴兴,
˜gT = 关g共0兲T关3兴,g共0兲T关4兴兴,
˜g共q * 兲 = 关g共q * 兲关5兴,g共q * 兲关6兴兴,
˜gT共q * 兲 = 关gT共q * 兲关3兴,gT共q * 兲关4兴兴.
The system ( ) has several solutions, (1) (2)
 = 0, h共1兲 = −␣共g共0兲T , r00兲 / zh0,  ⫽ 0, h共1兲 = −共g共0兲T , r02兲 / zh2, 共/␣兲2 = 共zho/zh2 · 共gT共q * 兲,r02兲 − 共g共0兲,r00兲兲/共g共0兲T,r22兲 = c.
If c ⬎ 0, then two nonhomogeneous solutions and one homogeneous solution 共 = 0兲 exist in vicinity h = h*, otherwise only homogeneous solution exists for  = 0. If hns ⬎ hst (or if hns does not exist), only the solution for  = 0 exists. From h共1兲 ⫽ 0 it follows that the bifurcation is transcritical. The condition h共1兲 = 0 in spinodal point corresponds to the critical point in Landau theory. VI. ANALYSIS OF THE STABILITY
The stability condition for the extremals is positive definiteness of the second variation of the free energy, which is the squared functional with respect to variation ␦⌽␣, and may be written as
␦ 2F = 1
冒 冕 冉兺 兺 V
␣

冊
共− ␦H␣ + ␦␣* 兲␦⌽ dx.
共40兲
The variations ␦H␣ are coupled with the variations ␦⌽␣ by the conditions (5) and (6),
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013301-11
Bifurcation analysis for the construction
J. Math. Phys. 46, 013301 (2005)
␦H␣ = T共− ␦⌽␣/⌽␣e + ␦U␣/U␣e + ␦V␣/V␣e 兲, ⵜ 2␦ U ␣ − ␦ U ␣ −
兺i Qi␣sei Uei /Vei ␦Vi = − 1/⌽0兺i Qi␣/Vei ␦⌽i ,
ⵜ 2␦ V ␣ − ␦ V ␣ −
兺i Q␣isei Vei /Uei ␦Ui = − 1/⌽0兺i Q␣i/Uei ␦⌽i .
共41兲
共42兲
The coefficients of the linear equations (41) and (42) are determined on the extremal 共Uei , Vei , sei ⌽ei 兲. It is necessary to convert the operator, which is defined by the left-hand sides of Eqs. (42) for obtaining the explicit form of the functional (40) via ␦⌽␣ ␦⌽. Because it is a rather unsolvable problem, for direct calculations we restrict ourselves by the set of periodic fluctuations ␦⌽␣, ␦⌽ with wave numbers q* and introduce Fourier expansions for ␦⌽␣, ␦⌽,
␦⌽␣ = 兺 ␦⌽共k␣c兲 cos共kq * x兲 + ␦⌽共k␣s兲 sin共kq * x兲. k
The system of Eqs. (41) and (42) is linear with respect to ␦U, ␦V, ␦⌽, therefore its solution can be presented in the form is 共is兲 ␦U␣ = 兺 共U␣ick共x兲␦⌽共ic兲 k + U␣k共x兲␦⌽k 兲,
共43兲
is 共is兲 ␦V␣ = 兺 共V␣ick共x兲␦⌽共ic兲 k + V␣k共x兲␦⌽k 兲,
共44兲
k,i
k,i
where the functions U␣ick共x兲, V␣ick共x兲 are solutions of the system of nonhomogeneous equations, ⵜ2␦U␣ick − ␦U␣ick −
兺j Q j␣sej Uej /Vej ␦Vicjk = − 1/⌽0Q␣i/Vei cos共kq * x兲,
ⵜ2␦V␣ick − ␦V␣ick −
兺j Q␣ jsej Vej /Uej ␦Uicjk = − 1/⌽0Q␣i/Uei cos共kq * x兲,
共45兲
and the functions U␣isk共x兲, V␣isk共x兲 are solutions of the same systems with substitution cos共kq * x兲 by sin共kq * x兲. Using the relations (41), (43), and (44), the functional (40) may be presented in the 共is兲 matrix form regarding ␦⌽共ic兲 k , ␦⌽k with elements, cs sc ss 2 2 2 共␦2F兲cc ij,kl,共␦ F兲ij,kl,共␦ F兲ij,kl,共␦ F兲ij,kl .
Their expressions are presented in Appendix B. Thus, the analysis of stability can be reduced to determination of the spectrum of matrix. When using km harmonics, the dimension of this matrix is m · 共2km + 1兲. VII. NUMERICAL CALCULATIONS
While calculating the phase diagrams, the basic difficulty is the solution of nonlinear systems (5), (6), and (10) for extremals. We used for this goal the iterated Newton method requiring a good starting approximation near bifurcation points if dealing with bifurcation analysis. Then we used a continuation procedure by parameter h with motion along tangent to curve z共h兲. Periodic boundary problem for linearized systems on every iteration was solved by the periodic sweep method.25 Similar procedure was used for the solution of the systems of equation for z / h by the method of tangents. The operator L determined by (28) has zero eigenvalue in the turn point of curve z共h兲 and method of tangents cannot be used. In this case a change of the sign of step for parameter h
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013301-12
Ivanova, Kuchanov, and Manevitch
J. Math. Phys. 46, 013301 (2005)
FIG. 1. Phase diagram for system (1). The curves (1), (2) are spinodal and binodal, respectively [for ⌽0 changing from 0 to 0.4 they are trivial ones, for ⌽0 ⬎ 0.4, curve 共1⬘兲, nontrivial spinodal], ⌽0 ⬵ 0.38 is the critical point (marked by 䊊), ⌽0 ⬵ 0.4 is the Lifshitz point (marked by ⫻).
and starting approximation were provided by extrapolation using two previous points. The calculations were performed for different copolymer systems and corresponding phase and bifurcation diagrams are presented in Figs. 1–5. 0.1 0 兲 Melt of diblock copolymers with structural matrix, Q = 共 0.9 0 0.95 0.05 , and matrix of interaction ␥ / ␥12共1 ; 1 ; 1.5兲. In this system only the trivial spinodal exists for values ⌽0 ⬍ 0.4, and both the trivial and nontrivial spinodals exist for ⌽0 ⬎ 0.4. The birth of nonhomogeneous structures in all points of nontrivial spinodals is subcritical by parameter h (and they are unstable). The starting appoximation for nonhomogeneous structures was obtained by formulas (38) and (39), and has continued by parameter h ⬍ h * 共⌽0兲 with help of the solutions of Eqs. (5), (6), and (10) up to the coups point h = ht. In this point the operator L共U , V , s , ⌽ , ht兲 has vanishing eigenvalue, and couple of structures (stable and unstable) appeared at h = h* on the nontrivial spinodal disappear. For h ⬎ ht the structures obtained by continuation procedure are stable up to value h = hb where R关⌽I兴 = 0. We calculate the spectrum of matrix (40) in every point of the curve z共h , ⌽0兲. Every branch z共h , ⌽0兲
FIG. 2. The spatial distribution ⌽2共x / l兲 for ⌽0 = 0.99 for the system (1) bold curve, for l = 260; next curve, for l = 2 / q * = 78.248 and value h, corresponding to binodal; dotted line, for h = hsn. The curves are plotted in the scale 共maxx ⌽2 − minx ⌽2兲.
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013301-13
Bifurcation analysis for the construction
J. Math. Phys. 46, 013301 (2005)
FIG. 3. Phase diagram for system (2). The curve (1), trivial spinodal, the curve (2) is nontrivial spinodal, (3) is the trivial binodal, ⌽0 ⬵ 0.28 is the Lifshitz point (marked by ⫻), ⌽0 ⬵ 0.099 is the critical point (marked by 䊊).
may be continued by parameter q (or l = 2 / q) for h ⫽ h * . The value of functional F关z兴 and magnitude hb decrease with increasing l. In these calculations the hb change was very small. It should be noted that the value hb for the homogeneous extremal arised transcritically in the spinodal point was practically the same as the value hb for the branch of nonhomogeneous struc0.01 0 兲 tures by l → ⬁. In Fig. 1 the phase diagrams are plotted, Q = 共 0.99 0 0.95 0.05 , and the matrix of interaction ␥ / ␥12共1.2; 1 ; 1.2兲. The phase and bifurcation (for ⌽0 = 0.5) diagrams for this system are shown in Figs. 3 and 4. For the nonhomogeneous structures ⌬⌽ = maxx ⌽2 − minx ⌽2, and for the homogeneous solution one deviation from the trivial one.
FIG. 4. The bifurcation diagram for the system (2) at ⌽0 = 0.5. The curve (1) is the unstable nonhomogenous branch, (2) is the stable one, (3) the stable part of the branch appeared transcritically from the spinodal point, (4) is the unstable part of the same branch, (5) is the stable homogeneous nontrivial branch rigidly arising in couple with branch (4).
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013301-14
Ivanova, Kuchanov, and Manevitch
J. Math. Phys. 46, 013301 (2005)
FIG. 5. The bifurcation diagram at ⌽0 = 0.76 for the system (3).
The nonlinear problem at hand can have far more complicated bifurcation diagrams than those described above. For example, let us consider the system, characterized by a set of input param0.15 0.84 0.01 兲 eters, Q = 共 0.84 0.15 0.01 , and the matrix of interaction ␥ / ␥12共0.08; 1 ; 1.5兲. The bifurcation behavior in the vicinity of the Lifshitz point (⌽0 = 0.745, h = 4.0857) is of particular interest from the physical standpoint. The periodic solutions of the set of equations (5), (6), and (10) have periods substantially exceeding the size of a monomeric unit. To carry out the bifurcation analysis with a single controlled parameter h we have chosen the value ⌽0 = 0.760 at which the above condition holds. For this value the bifurcation diagram is plotted in Fig. 5. As depicted in this picture the loss of the thermodynamic stability of the homogeneous state with the temperature decay is observed for the first time at point hsn = 4.3043 of the nontrivial spinodal. At this point the Turing subcritical bifurcation happens leading to the loss of stability of the nontrivial solution accompanied by the disappearance of the pair of unstable periodic solutions with q * = 0.5958. As h subsequently increases the instability interval of wave vectors ql ⬍ q ⬍ qr broadens up to the moment when h attains the value hst = 4.3185, corresponding to the trivial spinodal. At point h = hst, where ql = 0, qr = 0.986 the transcritical bifurcation takes place. Then the trivial extremal loses its stability with respect to homogeneous fluctuations whereas the upper part of nontrivial homogeneous extremal acquires it. Both of them, however, are unstable with respect to nonhomogeneous fluctuations in the vicinity of point h = hst. Interesting peculiarity of the bifurcation at this specific point is a splitting off from the nontrivial extremal of a pair of periodic ones corresponding to the wave vector q* unstable with respect to both types of fluctuations. The reason of such bifurcation behavior at point h = hst is that the zero eigenvalue of linear operator at this point has the multiplicity two, corresponding to the two-dimensional proper subspace (this bifurcation was described above). Apart from the primary bifurcations (points hsn, hst) in Fig. 5 some secondary ones (points h*, h * *) are also depicted which we managed to reveal numerically. When moving from point h = hst along the homogeneous nontrivial extremal the region of unstable wave vectors was found to reduce with the growth of control parameter h to contract to point q = q * * = 0.63 at h = h * * = 4.3365. As the result of this Turing supercritical bifurcation at point h * * the nontrivial extremal becomes absolutely stable giving birth to a couple of periodic structures with wave vector q * *. The free energy density of homogeneous nuclei described by this extremal turns out to be lower than that in the principal phase. When moving from point h = hst along the homogeneous nontrivial extremal in the direction of decreasing parameter h the wave vector value qr, bounding from the right-hand side of the interval of the instability of this extremal gets smaller to reach at point h* just that value q = q*, at which the stability loss by the trivial extremal happens at point hsn. The amplitude of the pair of periodic
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013301-15
J. Math. Phys. 46, 013301 (2005)
Bifurcation analysis for the construction
structures with wave vector q* emerging at point hsn as parameter h decreases, first increases to decay up to zero at the point when h = h*. At this point the Turing subcritical bifurcation occurs resulting in the disappearance of the pair of unstable periodic extremals. Simultaneously, the nontrivial homogeneous extremal, remaining in general unstable, acquires stability with respect to the fluctuations with q = q*. Under further decay of parameter h in the region to the left from the secondary bifurcation point the values ⌽1 and ⌽2 tend to zero at this extremal. Returning to the instability interval one can find, that in every point of the interval hsn ⬍ h ⬍ hst one can also observe the bifurcations of trivial solution. Type of these bifurcations will be different to the left and to the right of a certain specific point h− = 4.308, ql = 0.393 362, where qr = 2ql. To the left of this point both branches corresponding to ql and qr arise subcritically, each of them attaining the nontrivial homogeneous branch at corresponding points. To the right of point h− one branch remains subcritical, while the other becomes supercritical. Such a situation persists under changes of h up to a certain value h * * *, where coefficients h共2兲 and h共3兲 in the expansion (33) vanish. Subcritical branches reach at the bottom part of the nontrivial branch of the homogeneous extremal to the left of point hst, whereas the supercritical ones reach the upper part of this curve. When h ⬎ h * * * bifurcations of both branches again become subcritical and corresponding branches attain the bottom part of the nontrivial extremal. Specific point h− corresponds to the degenerate case since null space of the operator L at this point consists of two eigenvectors. Consequently the first approximation reads z共1兲 = ␣gl cos qlx + gr cos 2qlx. Constants ␣ and  are determined from the orthogonality conditions for two eigenfunctions gTl cos共qlx兲 and grT cos共2qlx兲 to the operator conjugate to L. The equations for determination of the ratio ␣ /  and h共1兲 are written as 1/2␣2共grT,B2共gl,gl兲兲 + h共1兲共grT,B2共gr,h兲兲 = 0, 1/2␣共gTl ,B2共gr,gl兲兲 + ␣h共1兲共gTl ,B2共gl,h兲兲 = 0,
共46兲
where B2共y1,y2兲 = 1/2
共2F/y 1i y 2j 兲y 1i y 2j , 兺 i,j
B2共y,h兲 =
兺j 共2F/y jh兲y j .
This system has several solutions (1) (2)
␣ = 0, h共1兲 = 0, ␣ ⫽ 0, h共1兲 = − / 2共gTl , B2共gr , gl兲兲 / 共gTl , B2共gl , h兲兲, 共␣/兲2 = 共gTl ,B2共gr,gl兲兲/共grT,B2共gl,gl兲兲 · 共grT,B2共gr,h兲/共gTl ,B2共gl,h兲兲 = c.
If c ⬎ 0 (in our case this condition is actually satisfied). The comparison of formulas (46) and (40) leads one to the conclusion that (with substitution ␣ by ) their second equations are identical while the first ones differ by the term proportional to 2 that is missing in set (46) by virtue of the orthogonality condition of functions cos共2qlx兲 and cos2共2qlx兲. The solution of set (46) (where ␣ = 0) describes the subcritical Turing bifurcation with two branches arising at the homogeneous nontrivial branch at h = h2ql. The second solution (where ␣ ⫽ 0) describes the transcritical bifurcation. Two unstable branches with period 2 / ql arise when h− = 2.24 at the nontrivial homogeneous extremal to the left of point h = hst, then one branch transcritically continues when h = hst in the direction of increasing of the parameter h to terminate at h = 4.33 at the homogeneous extremal. The bifurcation diagram presented in Fig. 5 is obviously incomplete. This statement is particularly evidenced by the absence in the interval hsn ⬍ h ⬍ h * * of extremals at which the Landau free energy functional has local minimum. Apparently, in this region of parameter h the nuclei of
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013301-16
J. Math. Phys. 46, 013301 (2005)
Ivanova, Kuchanov, and Manevitch
mesophases with two- or three-dimensional spatial periodicity must be stable. We have in mind to perform a theoretical analysis of the existence and stability of such nuclei in the subsequent publications.26–28 ACKNOWLEDGMENT
The authors gratefully acknowledge the financial support by CRDF (Grant No. RC-2-2398MO-02). APPENDIX A
The components of vectors r0 and rh in the formulas (39) are r0关i兴 = − 1/2
兺j ijg关j兴g关j + 2兴,
r0关i + 2兴 = − 1/共4兲共共g关i + 2兴/si兲2 − H1共g关5兴 + g关6兴兲2兲, r0关i + 4兴 = 1/2⌽0i共2Vig关i兴g关i + 2兴 + sig关i兴2兲, rh关i兴 = 0, rh关i + 2兴 = −
for i ⫽ 3,4,
兺j ␥ij共⌽0 j − ⌽ j兲,
where 共i = 1 , 2兲 the vector g is the eigenvector of the matrix J共q * 兲 and H1 = 1/共1 − ⌽1 − ⌽2兲2,
H2 = 2H1/共1 − ⌽1 − ⌽2兲.
The expression h共2兲 in the formulas (31) has the following form: h共2兲 = −兺i P共i兲 / 兺i Ph共i兲, where the denotations are accepted, ab共i兲 = a关i兴 + 1/2b关i兴,
ngT关i兴 = i1gT关1兴 + i2gT关2兴,
ahg关i兴 = g关i兴ah关i + 2兴 + ah关i兴g关i + 2兴, 2
P共1兲 = 1/2
兺1 ngT关i兴共ab共i兲g关i + 2兴 + ab共i + 2兲g关i兴兲,
P共2兲 = 1/2H1共gT关3兴 + gT关4兴兲共gT关5兴 + gT关6兴兲共ab共5兲 + ab共6兲兲, P共3兲 = 1/16H2共gT关3兴 + gT关4兴兲共g关5兴 + g关6兴兲3 , 2T
P共4兲 = − 1/2
兺1 g关i + 2兴g关i + 2兴/s2i 共ab共i + 2兲 − 1/共4Si兲g关i + 2兴2兲,
2
P共5兲 = − ⌽
0
兺1 Ti g关i + 4兴g关i兴共共3/8g关i + 2兴g关i兴 + ab共i + 2兲Vi兲 + ab共i兲共Vig关i + 2兴 + Sig关i兴兲兲,
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013301-17
J. Math. Phys. 46, 013301 (2005)
Bifurcation analysis for the construction 2
Ph共1兲 = 1/2
兺1 ngT关i兴ahg关i兴,
Ph共2兲 = H1/2共ah关5兴 + ah关6兴兲共gT关3兴 + gT关4兴兲共g关5兴 + g关6兴兲,
Ph共3兲 = 0,
2
Ph共4兲 = − 1/2
兺1 gT关i + 2兴共1/se2i g关i + 2兴ah共i + 2兲 − 共␥i1g关5兴 + ␥i2g关6兴兲兲, 2
Ph共5兲 = − ⌽ 共 0
兺1 igT关i + 4兴共共Sig关i兴 + Vig关i + 2兴兲ah共i兲 + ah共i + 2兲Vig关i兴兲兲.
Let n0共i兲 = 共i1g共0兲关1兴g共0兲关3兴 + i2g共0兲关2兴g共0兲关4兴兲, n2共i兲 = 共i1g共q * 兲关1兴g共q * 兲关3兴 + i2g共q * 兲关2兴g共q * 兲关4兴兲, VS共i兲 = 共2Vig共0兲关i兴g关i + 2兴 + si共g共0兲关i兴兲2兲, SS共i兲 = 1/2共共g共0兲关i + 2兴兲/si兲2 − H1共共g共0兲关5兴 + g共0兲关6兴兲2兲, 2
n02共i兲 =
兺1 ij共g共0兲关j + 2兴g共q * 兲关j兴 + g共0兲关j兴g共q * 兲关j + 2兴兲,
S02共i兲 = 1/2共g共0兲关i + 2兴g共q * 兲关i + 2兴/s2i − H1共g共0兲关5兴 + g共0兲关6兴兲共g共q * 兲关5兴 + g共q * 兲关6兴兲兲, VS02共i兲 = Vi共g共0兲关i + 2兴g共q * 兲关i兴 + g共0兲关i兴g共q * 兲关i + 2兴兲 + sig共0兲关i兴g共q * 兲关i兴, SS2共i兲 = 1/2共共g共q * 兲关i + 2兴/si兲2 − H1共g共q * 兲关5兴 + g共q * 兲关6兴兲2兲, VS2共i兲 = 2Vi共g共q * 兲关i + 2兴g共q * 兲关i兴兲 + Si共g共q * 兲关i兴兲2 . Then the vectors r00, r01, r22 have the following form: r00 = 关− n0共1兲,− n0共2兲,SS共1兲,SS共2兲,⌽01VS共1兲,⌽02VS共2兲兴, r02 = 关− 1/2n02共1兲,− 1/2n02共2兲,S02共1兲,S02共2兲,⌽01VS02共1兲,⌽02VS02共2兲兴, r22 = 关− 1/2n2共1兲,− 1/2n2共2兲,1/2SS共1兲,1/2SS共2兲,1/2⌽01VS共1兲,1/2⌽02VS共2兲兴. APPENDIX B
The matrix elements of the functional second variation are written as cc cc 共␦2F兲cc ij,kl = mij,kl + hij,kl,
cs cs 共␦2F兲cs ij,kl = mij,kl + hij,kl ,
sc sc 共␦2F兲sc ij,kl = mij,kl + hij,kl,
ss ss 共␦2F兲ss ij,kl = mij,kl + hij,kl ,
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013301-18
Ivanova, Kuchanov, and Manevitch
mcc ij,kl =
T L
冕
T L
冕
T L
冕
T L
冕
mcs ij,kl =
msc ij,kl =
mss ij,kl =
hcc ij,kl = −
hcs ij,kl = −
hsc ij,kl = −
hss ij,kl = −
T L
冕
T L
冕
T L
冕
T L
冕
J. Math. Phys. 46, 013301 (2005)
L
共− cij + 1/⌽i␦ij兲cos共kqx兲cos共lqx兲dx,
0
L
共− cij + 1/⌽i␦ij兲cos共kqx兲sin共lqx兲dx,
0
L
共− cij + 1/⌽i␦ij兲sin共kqx兲cos共lqx兲dx,
0
L
共− cij + 1/⌽i␦ij兲sin共kqx兲sin共lqx兲dx,
0
L
共1/Ui共x兲␦Uicj,k共x兲 + 1/Vi共x兲␦Vicj,k共x兲兲cos共lqx兲dx,
0
L
共1/Ui共x兲␦Uicj,k共x兲 + 1/Vi共x兲␦Vicj,k共x兲兲sin共lqx兲dx,
0
L
共1/Ui共x兲␦Uisj,k共x兲 + 1/Vi共x兲␦Visj,k共x兲兲cos共lqx兲dx,
0
L
共1/Ui共x兲␦Uisj,k共x兲 + 1/Vi共x兲␦Visj,k共x兲兲sin共lqx兲dx.
0
Here the values Ui共x兲, Vi共x兲 are given on the extremal, and ␦Uisj,k共x兲, ␦Uisj,k共x兲, ␦Uicj,k共x兲 ␦Vicj,k共x兲 are determined by Eq. (45), and L = 2 / q*. 1
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013301-19
Bifurcation analysis for the construction
J. Math. Phys. 46, 013301 (2005)
24
M. M. Vainberg and V. A. Trenogin, The Theory of Branched Solutions to Nonlinear Differential Equations (Nauka, Moscow, 1969). 25 A. A. Samarskii, Introduction to the Difference Schemes Theory (Nauka, Moscow, 1971). 26 Polymer Compatibility and Incompatibility, edited by K. Solk (Harwood Academic, London, 1982). 27 J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic, London, 1986). 28 J. P. Donley, J. G. Curro, and J. D. McCoy, J. Chem. Phys. 101, 3205 (1994).
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