J. Phys. Chem. B 2009, 113, 3853–3872
3853
Phase Diagrams of Semisoft Nematic Elastomers† Fangfu Ye*,‡ and T. C. Lubensky*,§ Liquid Crystal Institute, Kent State UniVersity, Kent, Ohio 44242-0001, and Department of Physics and Astronomy, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104 ReceiVed: September 15, 2008; ReVised Manuscript ReceiVed: December 18, 2008
Nematic elastomers with locked-in anisotropy direction exhibit semisoft elastic response characterized by a plateau in the stress-strain curve in which stress does not change with strain. We calculate the global phase diagram for a minimal model, which is equivalent to one describing a nematic in crossed electric and magnetic fields, and show that semisoft behavior is associated with a broken symmetry biaxial phase and that it persists well into the supercritical regime. We also consider generalizations beyond the minimal model, particularly the neo-classical model introduced by Warner, Bladon, and Terentjev, and find similar results. Our work underscores the fact that semisoft response is fundamentally a nonlinear phenomenon and that experiments on samples at zero external stress provide no information about it. I. Introduction 1
Nematic elastomers (NEs) are remarkable materials that combine the elastic properties of rubber with the orientational properties of nematic liquid crystals. It is appropriate for a journal issue honoring the late Nobelist P. G. de Gennes to have an article on liquid-crystal elastomers for he created the field. He published his first paper2 on the subject in 1969 in which he asked whether polymers dissolved and subsequently crosslinked in a liquid crystalline solvent could retain memory of the solvent’s orientational anisotropy once it was flushed from the now cross-linked network. His conclusion was that networks cross-linked in smectic or cholesteric liquid crystals would do so but those cross-linked in nematic liquid crystals would not. He later3,4 explored the interplay between nematic order and strain in elastomers formed via cross-linking nematic polymers,5 showing that strain could induce or rotate nematic order or vice versa; the use of nematic elastomers as artificial muscles6 and as actuators;7 and in a very late publication the nature of the phase transition from the isotropic to nematic phase and vice versa in a nematic rubber.8 This paper, which explores the global phase diagrams of nematic elastomers, including the region of soft9-11 and semisoft9,12-15 response is very much a descendant of de Gennes’s pioneering work. We are honored to have it appear in a journal volume dedicated to his memory. An ideal uniaxial nematic elastomer is produced when an isotropic rubber, formed by cross-linking polymers with nematogenic mesogens, undergoes a transition from the isotropic to the nematic phase in which it spontaneously stretches along one direction (which we take to be the z-direction) and contracts along the other two while its nematic mesogens align on average along the stretch direction. This ideal nematic phase exhibits what is called soft-elasticity9,10sa consequence of Goldstone modes arising from the breaking of the continuous rotational symmetry of the isotropic phase.11 Soft elasticity is characterized by the vanishing of the elastic modulus C5 measuring the energy associated with shear strains uxz and uyz in planes containing †
Part of the “PGG (Pierre-Gilles de Gennes) Memorial Issue”. * Towhomcorrespondenceshouldbeaddressed.E-mail:
[email protected];
[email protected]. ‡ Kent State. § University of Pennsylvania.
the anisotropy axis and by a stress-strain curve for strains uxx (or uyy) and stresses σxx (or σyy) perpendicular to the anisotropy axis in which strains up to a critical value can be produced by zero stress. Monodomain samples cannot be produced without locking inapreferredanisotropydirection,usuallybytheKu¨pfer-Finkelmann (KF) procedure16 in which a first cross-linking in the absence of uniaxial stress is followed by second one with stress. This process introduces a mechanical aligning field h, analogous to an external electric or magnetic field, and lifts the value of the elastic modulus C5 from zero. Thus, nematic elastomers prepared in this way are simply uniaxial solids with a linear stress strain relation at small strain. Their behavior in response to stress in the geometry that produces soft response when h ) 0 is particularly interesting. For large h, they should exhibit a stress-strain curve in which σxx is a continuous monotonically increasing function of uxx. However, for fields h that are not too large they are predicted1,13 and indeed are observed12,15 to exhibit semisoft response to an external stress σxx. As depicted schematically in Figure 1, the curve of σxx as a function of uxx rises linearly at small uxx as it would in a uniaxial solid. However, at a critical value of uxx ) uxx at which point σxx ) h, the slope of σxx versus uxx abruptly decreases to zero giving rise to a flat plateau that persists to a second critical strain uxx ) u+xx beyond which the stress rises again with strain. Throughout the semisoft plateau, the shear strain uxz is nonzero: it can have either a positive or a negative sign, but its magnitude rises from zero at uxx ) uxx, reaches a maximum, then vanishes at uxx ) u+ xx. Thus, the plateau is intimately connected to a phase transition, induced by external stress, in which uxz spontaneously develops a nonzero value. In a theoretical idealization, monodomain samples with a spatially uniform uxz of a single sign form when there is a semisoft plateau. In real systems, however, boundary conditions enforce the formation of textures in which regions of opposite signs of uxz separated by domain walls coexist.13,17-19 The region in the phase diagram exhibiting semisoft response, which exists only on the plane σxx ) h (and its symmetry equivalents), is a separate thermodynamic phase, which we call the semisoft phase. It is characterized by biaxial order in the strain tensor _u that defines a preferred direction, which we call
10.1021/jp8082002 CCC: $40.75 2009 American Chemical Society Published on Web 03/19/2009
3854 J. Phys. Chem. B, Vol. 113, No. 12, 2009
Figure 1. Schematic representations of different stress-strain responses: (a) soft response of the second PK stress σxx to uxx, (b) semisoft response of σxx to uxx, and (c) semisoft response of the first PK stress σIxx to Λxx. Panel b provides schematics of the sample morphology in the semisoft region between u-xx and u+xx, and in the normal elastic regions + uxx < uxx and uxx > uxx. The double arrows in these figures indicate the direction of the principal strain anisotropy axis. In the semisoft regime, samples adopt a polydomain structure with coexisting regions with opposite signs for uxz. The semisoft plateau in panel b is flat, whereas that in panel c rises linearly with Λxx. In both cases, samples exhibit polydomain morphology.
the biaxial anisotropy direction, in xz-plane perpendicular to the high-temperature direction of uniaxial order along the y-axis. The direction of biaxial anisotropy in the xz plane is arbitrary in the absence of externally imposed stress or strain (other than σxx). It is this soft biaxial order that is ultimately responsible for the semisoft response. When σxx ) h-, then _u is diagonal with its uniaxial axis along y and its biaxial anisotropy axis along z. As uxx is increased beyond uxx, the direction of biaxial anisotropy rotates in the xz plane creating a nonzero uxz while preserving the magnitude of the biaxial order parameter and thus not changing the energy of the system. σxx remains fixed and equal to h during this rotation. This description of semisoft response appears to be general and to apply to all models of this phase when expressed in terms of appropriate parameters. The frozen-in anisotropy direction (along z and defined by the field h in the preceding discussion) and σxx define the soft plane and the oblate uniaxial direction perpendicular to it. Straininduced rotations in the soft plane are responsible for semisoft response. In soft rather than semisoft systems, order is prolate uniaxial, and strain-induced soft rotations (which simply rotate the direction of the uniaxial axis) can take place in any plane containing the initial uniaxial anisotropy axis. Semisoft elasticity is not the only mechanism that can produce a flat plateau in the stress-strain curve. If two distinct and discrete values of uxx (say uxx1 and uxx2) can coexist at a given value of σxx, then the ideal stress-strain curve will exhibit a flat plateau for uxx1 < uxx < uxx2 just as the pressure-density curve in a classical fluid exhibits a flat plateau in the region of liquid-gas coexistence. The coexistence of two or more discrete states with different shape anisotropies is a characteristic of Martensites20-crystals that undergo a phase transition, for example from cubic to tetragonal symmetry, and we will refer to the phases with such coexistence as Martensitic phases. In the case of liquid-gas coexistence, there is in equilibrium a single domain wall separating the liquid and gas phases, and the fraction of the sample in the liquid phase increases as the
Ye and Lubensky
Figure 2. (a and b): Regions in the r-h plane of the minimal model exhibiting different responses to second PK stress σxx. The region of semisoft response (S-S) is indicated with horizontal dashed lines and that of smooth response is indicated with no marking. The region in which there are two instances of Martensitic response upon increasing σxx is indicated with vertical lines and that with one instance is indicated with dots. (a) Small r and h: showing the PN-N coexistence line terminating at the mechanical critical point cZ. (b) Same figure on a larger scale showing the large region of semisoft response. Panel a is a blow-up of the region marked with a (red) rectangle in panel b. Panel c shows similar response to a first PK stress. There is a region of semisoft response (S-S, horizontal lines), a region of Martensitic response (dotted) that is more complex [see section IIIF] than that arising from second PK stress, and a region of smooth response on increasing σIxx from zero.
density increases. In the elastomer case single domain walls can occur as in necking (Considere instability) of polymers under stress.1,21 More generally, more complicated domain-wall structures may be required to meet imposed boundary conditions. These observations lead to the conclusion that the stress-strain response of nematic elastomers can exhibit a rich variety of behaviors. Depending on the values of temperature, which we measure by a parameter r, and the internal aligning field h, a nematic elastomer could exhibit a smooth, a soft, a semisoft, or a Martensitic stress-strain curve. Our principal goal in this paper is to identify those initial conditions (i.e., initial values of r and h) that will exhibit semisoft response. Smooth response occurs when there is no phase transition when σxx is increased; soft response occurs only when h is zero and r is low enough that the equilibrium system is in the nematic rather than the isotropic phase; and both Martensitic and semisoft responses occur when a phase transition occurs upon increasing σxx. Thus, the global phase diagram in the space defined by r, h, and σxx contains full information about the conditions under which semisoft and other responses occur. In this paper, we address the conditions that lead to semisoft response in a number of models. We focus mostly on a simplified model, which we call the minimal model, because most of its global phase diagram and associated stress-strain curve can be calculated analytically. In this model, which is formally equivalent to that of a nematic liquid crystal in orthogonal electric and magnetic fields, the order parameter is the Green-Saint-Venant strain tensor _u22-24 with components uij with the constraint Tru_ ) 0 [rather than the true incompressible constraint det(1 + 2u _) ) 0], and the external stress σij ) ∂f/∂uij is the second Piola-Kirchhoff (PK)25 or thermodynamic stress tensor,25 where f is the free energy density. Figure 2 shows the phase diagram of this model projected onto the r-h plane. This diagram shows the familiar line of coexistence of a paranematic (PN) phase with a nematic (N) phase terminating
Semisoft Elastomers at a mechanical critical point3 at (r, h) ) (rc, hc). It also shows the projection onto the r-h plane of semisoft phase (S-S), which exhibits semisoft response, and the two-phase coexistence region, which exhibits Martensitic response, upon increasing σxx. It clearly shows that the region of semisoft response exists for values of r and h, respectively greater than rc and hc and at sufficiently small r for any given h. Interestingly a sample near the mechanical critical point will exhibit Martensitic rather than semisoft response. The qualitative features of this phase diagram do not change when the constraint that Tru _ ) 0 is replaced with the constant volume constraint det(1 + 2u _) ) 1. The second PK stress tensor is not strictly speaking a stress tensor, that is, it is not a physical force per unit area (though it has these units), and it cannot be continuously changed by external forces. Generally in experiments such as those that probe semisoft response in elastomers, it is the first PK or _ is the Cauchy engineering stress tensor, σIij ) ∂f/∂Λij, where Λ deformation tensor, that is externally controlled. The strain _ TΛ _ -δ _). Thus, for these tensor is related to Λ _ via _u ) 1/2(Λ experiments, we should consider the σIxx-Λxx rather than the σxx-uxx stress-strain curve. For a given free energy density f that depends only on _u, σIxx ) Λxxσxx. Thus, the flat plateau in the σxx-uxx curve slopes slightly upward in the σIxx-Λxx curve as shown in Figure 1c. This means that there is a unique value of σIxx for each Λxx and, as a result, there are important differences between the r-h-σIxx and the r-h-σxx phase diagrams that, however, do not qualitatively modify the geometry of the regions in the r-h subspace that exhibit semisoft and Martensitic response. The minimal model and its modification to impose a true incompressible constraint or to replace σxx by σIxx are purely phenomenological, and they depend only on strain but not on truenematicorientationalordermeasuredbytheMaier-Saupe-deGennes order parameter Qij or the Frank director n. Verwey and Warner14 developed a semimicroscopic model, based on the very successful neoclassical model for nematic elastomers, for systems with semisoft response. In this model, there are compositional fluctuations along chain segments characterized by a parameter R with mean 〈R〉 and variance 〈(δR)2〉, and the analogue of the anisotropy field h is a combination of the nematic order parameter Q0 at the time of the second crosslinking and 〈(δR)2〉. We investigate regions of semisoft response in this model and establish that this response exists well away from the critical point. However, because this model is characterized by at least four parameters (r, Q0, 〈(δR)2〉, σIxx), we do not attempt to calculate a global phase diagram. The Goldstone arguments11 for soft-response predicts that C5 ) 0 in the nematic phase, making reasonable conjectures that C5 should remain small at finite h when semisoft response is expected and that semisoft response might not exist at all in the supercritical regime26 beyond the mechanical critical point (with h ) hc) terminating the PN-N coexistence line.3 There is now strong evidence27,28 that samples prepared with the KF technique are supercritical. In addition, C5 measured in linearized rheological experiments is not particularly small.28 These results have caused some to doubt the interpretation of the measured stress-strain plateau in terms of semisoft response.29 The purpose of this paper is to clarify the nature of semisoft response and provide a complete phase diagram for nematic elastomers subjected to aligning or stress fields h and σxx along orthogonal directions. We show in particular that linearized rheology measurements at zero stress provide no information about semisoft response. We first introduce in section II some preliminaries needed in the following sections. In section III,
J. Phys. Chem. B, Vol. 113, No. 12, 2009 3855 we explore the simplest model, which we call the minimal model, that exhibits semisoft response. This model is formally equivalent to the Maier-Saupe-de-Gennes model30 for a nematic in orthogonal electric and magnetic fields.31 We derive the global mean-field phase diagram for this model and show that semisoft response is associated with biaxial phases that spontaneously break rotational symmetry. We discuss phase diagrams for both second and first PK stresses. We briefly discuss other models in section IV, including one that generalizes the minimal model to include a true incompressible rather than a traceless constraint on the strain and one with a nonlinear rather than a linear anisotropy energy. In section V, we study the semisoft version14 of the neoclassical model.1,32 Finally, in section VI, we present a brief summary of our results. In addition, in a series of four appendices, we discuss subtleties associated with the traceless constraint and present various algebraic details. II. Preliminaries A. Strain and Elastic Free Energy. An elastomer is characterized by a reference configuration, which we refer to as a reference space, SR, with mass points at positions x. Upon distortion of the elastomer, points x are mapped to points R(x) ≡ x + u(x) in a target space ST, where u is the displacement variable. Elastic distortions that vary slowly on scales set by the distance between cross-links are described by the Cauchy deformation tensor Λ _ with components Λij ) ∂Ri/∂xj. The usual Green-Saint-Laurent strain tensor is then
1 T u_ ) (Λ _ -δ _) _ Λ 2
(1)
_ and δ _ is the unit tensor. _u is where Λ _ T is the transpose of Λ invariant under arbitrary rotations of the target space and transforms like a rank-2 tensor under the rotations of reference space. The orientational properties of nematic mesogens in the elastomer are measured by the usual Maier-Saupe nematic tensor Q. A complete theory for nematic elastomers should treat both Λ _ and Q and couplings between them. We are, however, principally interested in the mechanical properties of these elastomers, and an effective model free energy, obtained by integrating out Qij;33 that is, a function of u_ only will be sufficient for most of our purposes.11,35 Furthermore, we will assume in such a theory that strains measure distortions relative to an isotropic reference state, which we take to be the infinitetemperature equilibrium state of the system after the second cross-linking. Thus the elastic free-energy per unit volume of the reference space f(u_) consists of an isotropic part fiso(u_), which is invariant under separate rotations in the reference and target space, and an anisotropic part fani(u_,h), invariant under rotation in the target but not the reference space, arising from the anisotropic imprinting process,16
f(u_) ) fiso(u _) + fani(u _,h)
(2)
where h is the internal aligning field introduced during the second cross-linking, which we take to favor alignment along the z-direction. B. Stress and Gibbs Energy. Stresses are related to derivatives of f with respect to Λ _ or _u. The internal force on a small volume element centered at x arises from forces exerted across its surfaces by the medium surrounding it. Thus, the force per
3856 J. Phys. Chem. B, Vol. 113, No. 12, 2009
Ye and Lubensky
unit volume in reference space, f, must be the divergence of a tensor:
With stresses given, the energy whose minimization determines equilibrium strains is
∂σijI fi ) ∂xj
g(u _) ) f(u _) + fext
(3)
σIij 25
Here, is the first Piola-Kirchhoff (PK) or engineering stress tensor. It is the force in direction i per unit area in the reference material directed along a surface element in that material with normal along j. The i and j indices transform, respectively, under operations in the target and reference spaces, and σIij is a mixed tensor that is not in general symmetric. σIij measures internal forces exerted by the medium: it is nonzero only in the interior D of a sample, and it must be zero on the boundary ∂D of the sample. The change in free energy arising from work done by internal forces displacing mass points from R(x) to R(x) + δR(x) is
δF ) -
∫D d
x fiδRi(x) ) -
d
)
∫D
∂σijI d x δR (x) ∂xj i d
∫D d dxσijIδΛij(x)
(4)
(5)
where the integral over the surface ∂D arising from integration by parts vanishes because σIij is zero on ∂D. Thus,
σijI )
δF ∂f ) ≡ Λikσkj δΛij ∂Λij
(6)
(10)
where fext is -σxxuxx for the second PK stress or -σIxxΛxx for the first PK stress. The subscript “ext” denotes that this term is induced by the external force pulling along x-direction in the case of the first PK stress. The equilibrium state can then be determined by minimizing g(u_) over _u for the second PK stress and over Λ _ for the first PK stress. We will often refer to the function g(u_) as the Gibbs energy density even though strictly speaking minimization of this energy over _u is required to produce the actual Gibbs energy. C. Transformation Between Two Reference States. The strain and stress tensors used in eq 2 and eq 10 are defined with respect to a nominal isotropic reference state. However, in most mechanical experiments, strains and stresses are measured with respect to the equilibrium state at any given temperature, which is normally different from the reference state we use in the strain-only theory. It is, therefore, necessary to obtain the transformation relation between these two sets of variables. The new equilibrium reference state with nematic order is a _ 0TΛ _ 0 - δ)/2 associated with uniaxial state with strain _u0 ) (Λ the target-space position R0i (x) ) Λ0ijxj ≡ xi′. The deformation tensor Λij ) ∂Ri/∂xj relative to the original isotropic reference state is related to that Λij′ ) ∂Ri/∂xj′ relative to the new referencespace positions xi′ via
Λ _ )Λ _ Λ _0
(11)
The strain _u′ relative to the new space is thus
where
σkj )
∂f ∂ukj
(7)
is the second PK stress tensor, which is symmetric, invariant under rotations in the target space, and a second-rank tensor with respect to rotations in the reference space. The usual textbook stress for physicists is the Cauchy stress, σCij , the force per unit area in the target space. To obtain the Cauchy stress in terms of the Piola-Kirchhoff stresses, we need to express all fields as a function of R rather than x. Thus,
∆F ) )
∂δR
∫ d dxσijI ∂xj i
)-
∫
∂σilC d R δR ∂Rl i
(8)
σijC )
1 I 1 σilΛjl ) Λ σ ΛT det Λ _ det Λ _ ik kl lj
(9)
σCij is thus symmetric as the conservation of angular momentum requires.
(13)
The first PK stress tensor relative to the new reference state ) (detΛ _ 0)-1(∂Λkl/∂Λ′)(∂f/∂Λ ) is σ′ijI ) (∂f ′/∂Λ′) ij ij kl) 0 -1 I 0 (detΛ _ ) σikΛ jk, namely,
σ _I ) (det Λ _ 0)-1σ _IΛ _ 0T
We, therefore, have
(12)
where δu_ ) u_ - u_0. The strain _u′, like _u, is invariant with respect to rigid rotations in the target space, and the elastic energy of broken-symmetry phases can be expressed as a function of it alone rather than as a function of Λ _ ′. The second PK stress tensor measured with respect to the _ 0)-1f is new reference state is σij′ ) ∂f ′/∂uij′, where f ′ ) (detΛ the free energy density in the new reference state. The relation between σij′ and σij is determined by the requirement that _δu_ ) ∫d3x′Trσ _′δu_′ or ∫d3xTrσ
σ _ ) (det Λ _ 0)-1Λ _ 0σ _Λ _ 0T
∂δR
∫ d dR det1 Λ_ σilI Λjl ∂Rj i d
1 T _u ) (Λ _ 0T)-1δu_(Λ _ 0)-1 ) (Λ _ -δ _) _ Λ 2
(14)
The transformation relations established above allow us to safely use variables defined in the nominal isotropic reference state and enable us to utilize the rotational symmetry associated with this isotropic state. Therefore, in the following sections, except section V, which treats the neoclassical model, only
Semisoft Elastomers
J. Phys. Chem. B, Vol. 113, No. 12, 2009 3857
energies expressed in terms of variables defined in this isotropic reference state will be analyzed. III. Minimal Model Semisoft elasticity manifests itself at finite values of strain or stress and is thus fundamentally a nonlinear phenomenon. The semisoft phase exists in the case of an imposed second PK stress σxx in a region of the global phase diagram in the spaces defined by r, h, and σxx or by r, h, and uxx. In this section, we will analyze full phase diagrams in the minimial model, which we also introduce and define. We will also explore how spontaneous biaxial order, the distinguishing property of the semisoft phase, gives rise to a flat stress-strain curve. The shear moduli of nematic elastomers, which are set by weak entropic polymer stretching forces and cross-link density, are orders of magnitude smaller than the bulk modulus, which is set by short-range enthalpic interactions between monomers. As a result, nematic elastomers are essentially incompressible. _ is the This means that detΛ _ ) [det(δ _ + 2u _)]1/2 ) 1, where δ unit matrix. Implementation of this constraint introduces computational complexity, and in our minimal model, we will replace it with the simpler constraint Tru _ ) 0, which enforces the true incompressible constraint at small strain but not at large strain (subtleties, associated with the constraint Tru _ ) 0 will be discussed in App. A). We will return later to the correct constraint. In addition, we find it convenient and instructive to work first with the second PK stress, σxx, tensor though most experiments, which apply a controlled force, produce the engineering stress, σxxI . (We will return to the first PK stress at the end of this section.) Furthermore, in our minimal model, the anisotropic part of the free energy fani is assumed to be a linear function of uzz:
fani ) -huzz
(15)
With these assumptions, our minimal model becomes formally equivalent to that for a nematic liquid crystal in crossed electric and magnetic fields, E ) Eez and H ) Hex, in which uij T Qij, h T 1/2∆εE2, and σxx T 1/2χaH2, where ∆ε and χa are, respectively, the anisotropic parts of the dielectric tensor and the magnetic susceptibility, and ea, a ) x,y,z, are unit vectors along direction a. (Note, however, that uij transforms like a tensor in the reference space, whereas Qij is normally taken to transform like a tensor in the target space.) The Gibbs free-energy density of the minimal model is thus
gm(u _,h, σxx) ) fiso(u _) - huzz - σxxuxx
(16)
fiso is chosen to have the Landau-de-Gennes form,30
1 fiso(u _) ) rTru _3 + V(Tru _2)2 _2 - wTru 2
(17)
where we assume w > 0 and where r ) a(T - T0) with T the temperature and T0 is the mean-field limit of metastability of the isotropic phase when h ) 0 and σxx ) 0. Note that fiso(u_) is invariant with respect to rotations in the reference space as well as in the target space. In the isotropic phase with _u ) 0, we have r ) 2µ, where µ is the T-dependent shear modulus. As in a nematic liquid crystal, the minimum energy state for fiso(u_) exhibits spontaneous symmetry breaking with a uniaxial order
parameter (u _) below the isotropic-to-nematic transition at r ) rN ) 1/12w2/V. To reduce the number of parameters, we will express quantities in reduced form: g˜ ) (V3/w4) · g, u_˜ ) (V/w) · u_, r˜ ) _˜ ) (V2/w3) · σ _, C˜5 ) (V/w2) · C5, (V/w2) · r, h˜ ) (V2/w3) · h, and σ and similarly for other elastic moduli. Thus for the minimal model we have
1 g˜ ) r˜Tru _˜ 3 + (Tru _˜ 2)2 - h˜u_˜ zz - σ˜ xx_u˜xx (18) _˜ 2 - Tru 2 We will drop the ∼ sign in most parts of this section to avoid tedious notation. Therefore, unless otherwise indicated, the quantities appearing in this section actually represent their reduced form though there is no ∼. In the remainder of this section, we will calculate in section IIIA the phase diagram in the r-h plane for both positive and negative h when no external stress σxx is applied. The h < 0 part of this plane, as we shall show, is equivalent by symmetry to the plane σxx ) h with h > 0 on which the semisoft phase exists. Thus, determination of its properties is a valuable first step toward understanding the global three-dimensional phase diagram. We then deduce in section IIIB the full phase diagram in the r-h-σxx space and in section IIIC the phase diagram in the r-h-uxx space. In section IIID, we derive stress-strain curves and show how semisoft response is related to coexistence of biaxial states on the vertical σxx ) h plane. In section IIIE, we derive general properties of stress-strain curves from Ward identities. In section IIIF, we consider an externally imposed first PK stress σxxI rather than a second PK stress σxx. Finally in section IIIG, we discuss strains relative to the physical initial state rather than the isotropic but not physically accessible reference state. The region of the r-h-σxx phase space of primary physical interest is that with h > 0 and σxx > 0. We consider, however, the full phase diagram for both signs of h and σxx to highlight symmetry properties such as the equivalence of the h < 0 part of the σxx ) 0 plane with the h > 0 part of the σxx ) h plane. Of course, our full phase diagram does allow us to make predictions about the behavior of a material with negative (oblate) uniaxial anisotropy frozen in by a negative h in response to a stress σxx or the response to a negative σxx when h > 0 as we discuss briefly in section IIID. A. Phase Diagram on the σxx ) 0 Plane. We begin our analysis of the global phase diagram31,36 with the σxx ) 0 plane, which we will refer to as the Z-plane because the anisotropy field h favors uniaxial order along the z-axis. We will introduce shortly X- and Y-planes in which anisotropy fields defined by h and σxx favor uniaxial order along the x- and y-axes, respectively. (The Y-plane is the one responsible, as we shall see, for semisoft response.) When h ) 0, there is the standard isotropic-to-nematic phase transition, which occurs at r ) rN ) 1/12. The resulting nematic phase is prolate uniaxial with order parameter uij ) S(ninj - 1/3δij), where S > 0 and the unit director n can point anywhere on the unit sphere. When h > 0, uij is prolate uniaxial at all temperature, and n ) ez, the unit vector along the z-axis of the reference space. When h < 0, there is oblate uniaxial order with S < 0 and n ) ez at high temperature. At low temperature, where nematic order exists when h ) 0, the initial effect of turning on a small h < 0 is to align n in the plane perpendicular to ez. This creates a biaxial environment with distinct orthogonal directions n and ez ⊥ n and causes development of biaxial order. Thus the appropriate form of the order parameter in the Z-plane is
3858 J. Phys. Chem. B, Vol. 113, No. 12, 2009
1 uij ) S eziezj - δij + η1(exiexj - eyieyj) + 3 η2(exieyj + eyiexj) 1 ) S eziezj - δij + η(e1ie1j - e2ie2j) 3 S η2 - + η1 0 3 S η2 - - η1 0 ) 3 2S 0 0 3
(
(
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Ye and Lubensky
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)
g)r
(19)
The vector e1 ) (cos θ, (sin θ, 0) specifies the principal axis of biaxial order, e2 ) ez × e1 ) (-sin θ, cos θ, 0), η1 ) η cos 2θ, η2 ) -η sin 2θ, and η2 ) η12 + η22. The full range of possible order parameters is described by either sign of the sin θ terms in e1 and e2 and associated sign of sin 2θ in η2 for 0 < θ < π. To show explicitly the symmetry between +uxz and -uxz, we will describe that full range of order parameters by restricting θ to lie between 0 and π/2 but by using both signs η ) (η1,η2) can in the definition of e1 and consequently of η2. b be viewed as a two-dimensional biaxial order parameter. To keep our discussion as simple as possible, we do not at this point include uxz and uyz contributions to _u. These strains, which are critical to the phenomenon of semisoftness, are in fact zero in equilibrium until σxx ) h, and they can be ignored in our current analysis of phase behavior in the vicinity of the Z-plane. The representation of uij in eq 19 allows us to describe all of the states of interest to us. Since _u is traceless, it must have at least two distinct eigenvalues. If it has two, it is uniaxial, and if it has three, it is biaxial. A uniaxial tensor is either prolate or oblate depending on whether its largest magnitude eigenvalue is positive or negative. The direction of uniaxial anisotropy is the direction associated with the largest magnitude eigenvalue. Thus, when η ) 0, _u of eq 19 is prolate for S > 0 and oblate for S < 0, and in both cases, the direction of uniaxial anisotropy is along ez. When η g 0 increases from zero, _u develops three distinct eigenvalues, and it becomes biaxial and develops anisotropy in the xy plane. We will call the direction associated with the largest eigenvalue in the xy plane the direction of biaxial anisotropy and the z direction the direction of uniaxial anisotropy (defined by the eigenvalue of maximum magnitude when η ) 0). Thus, when S < 0 the direction of biaxial anisotropy is along e1, that is, along ex when θ ) 0 and ey when θ ) π/2). There is an alternative convention, which is often used, that associates the direction of uniaxial anisotropy with the largest-magnitude eigenvalue and the direction of biaxial anisotropy with the direction of the eigenvalue of second-largest magnitude. This convention is inconvenient for our purposes because it would require us to change the names of the uniaxial and biaxial directions when the relative magnitude of eigenvalues interchange. For example, when η becomes greater than |S|/3 when S < 0, the eigenvalue of largest magnitude is (|S|/3) + η > 0 and that of second largest magnitude is 2S/3 < 0, and the uniaxial and biaxial directions would, respectively, be along e1 and ez rather than continuing to be along ez and e1 as they are in our convention. It is worth observing that the maximum-magnitude eigenvalue passes from negative (2S/3) to positive (1/3 |S| + η) in this transition. If η ) |S|, u_ becomes uniaxial. We will find that this limit is not reached in the semisoft phase. The Gibbs free energy density associated with strain tensor of eq 19 is
(
)
(
)
S2 S2 2 + η2 - S3 + 2Sη2 + 4 + η2 3 9 3 2 1 hS + σxxS - σxxη1 3 3
2
-
(20)
Note that when σxx ) 0, g depends only on the rotationally invariant form of the biaxial order parameter, η2 ) η12 + η22. The equations of state for the order parameters S, η1, and η2 are
3 ∂g 8 1 ) rS - S2 + S3 + (3 + 8S)η2 - h + σxx ) 0 2 ∂S 3 2 (21a) 3 ∂g 8 3 ) 3η1 r + 2S + S2 + 8η2 - σxx ) 0 2 ∂η1 3 2
(21b)
3 ∂g 8 ) 3η2 r + 2S + S2 + 8η2 ) 0 2 ∂η2 3
(21c)
( (
) )
If σxx ) 0 and h > 0, then S > 0 and r + 2S + 8/3S2 > 0, and the only solution to these equations is the uniaxial one with η1 ) η2 ) 0. Thus, for h > 0, the equation of state on the Z-plane for S is eq 21a with η ) 0 and σxx ) 0. At large r, there is only one real solution to this equation that approaches S ) h/r as r f ∞. At h ) 0, the isotropic phase with S ) 0 and the nematic phase with S ) SN ) 1/4 coexist at r ) rN ) 1/12, and for h > 0, there is a line of coexistence of the uniaxial paranematic (PN) phase and the uniaxial nematic (N) phase terminating at a liquid-gas-like critical point cZ at (r, h) ) (rc, hc) and S ) Sc as shown in Figure 3a. The order parameter S satisfies the equation of state
3 ∂g 8 ) rS - S2 + S3 - h ) 0 2 ∂S 3
(22)
in all uniaxial phases. We will denote these uniaxial solutions as S ) Su(r, h). The three critical-point parameters rc, hc, and Sc are determined by the equation of state, eq 22, and the two additional conditions ∂2g/∂S2 ) 0 and ∂3g/∂S3 ) 0. Expressing g in terms of δS ) S - Sc, we obtain
Figure 3. Phase diagram on the σxx ) 0 plane. The area inside the (red) dashed rectangle in panel a is shown in panel b. The line rB(|h|) is the phase boundary between the high-temperature phase and the lowtemperature phase. The thick part of rB(|h|) represents the first-order phase boundary and the thin part represents the second-order one. The dashed region SZ is the biaxial region. The thick dashed line in panel b is the paranematic-nematic coexistence line. The tricritical point (tZ), the nematic clearing point (Nc) and the mechanical critical point (cZ) are also shown in panel b.
Semisoft Elastomers
J. Phys. Chem. B, Vol. 113, No. 12, 2009 3859
3 1 2 1 g ) rS2c - S2c + S4c 2 2 3 3 8 2 hSc rSc - Sc + S3c - h δS + 3 2 8 1 1 r - Sc + 4S2c δS2 - Sc δS3 + δS4 (23) 2 3 3 3
(
)
(
)
(
)
δg ) Cηη2 + CSδS2 + 2CSηδSη2 2 2 1 [1 - 8Su(r, h)]δS3 + 4 η2 + δS2 (31) 9 3
Thus, at the critical point, S satisfies eq 22 and
3 ∂2g ) r - 2S + 8S2 ) 0 2 ∂S2
(
(24)
3 ∂3g ) 16S - 2 ) 0 2 ∂S3
in S to obtain a free energy that is a function of η alone. In the high-temperature phase, S ) Su(r, h) satisfies the equation of state, eq 22. Defining δS ) S - Su(r, h), and expanding δg ) g(Su(r, h) + δS,η) - g(Su(r, h), 0) in powers of δS and η, we obtain
where
(25)
Solving these equations, we obtain
1 rc ) , 8
hc )
1 , 192
Sc )
1 8
3 1 2 3 g ) gc(∆r, ∆h) + ∆rδS2 + δS4 - (∆h - ∆rSc)δS 2 2 2 3 (27) where gc(∆r,∆h) ) -1/(4×83) + 1/2(∆rSc - 2∆h)Sc. Note that there is no δS3 term in eq 27. This is because the third-order term in eq 23 depends only on Sc and not on h or r. The coexistence line is the line along which the coefficient of δS in eq 27 vanishes, that is,
(h - hc) ) (r - rc)Sc
(28)
The Gibbs free energy, gc, at S ) SC contributes to the entropy and specific heat, but it does not determine the order parameter, which along the coexistence line is
3 δS ) ( |∆r| 8
(
1/2
)
8 Cη ) r + 2Su(r, h) + S2u(r, h) 3
(32a)
1 CS ) [r - 2Su(r, h) + 8S2u(r, h)] 3
(32b)
8 CSη ) 1 + Su(r, h) 3
(32c)
(26)
Reexpressing g in terms of ∆r ) r - rc and ∆h ) h - hc, we obtain
Integrating out δS from eq 31 yields
δg ) Cηη2 + Dηη4 + O[η6]
3 1 ( |∆r| 8 8
(
(33)
where
(
)
2 CSη 4 3 8 Dη ) 4 ) r - - 6Su(r, h) + S2u(r, h) CS 3CS 4 3 (34)
(
)
Equation 33 is a standard Landau free energy for a system with O2 symmetry. When Dη > 0, there is a second-order transition at Cη ) 0. This relation, along with the equation of state for Su(r, h) (eq 22), determines the equation for the second-order line (for |h| > |ht|),
|h|3 - 98 |h| ) 0
rB(h) ) 2
(29)
(35)
and the order parameter,
Thus, along the coexistence line, the full nematic order parameter is
S)
)
1/2
)
(30)
As we discussed earlier, when h < 0, there is a transition from an oblate uniaxial state to a biaxial state, which exists on a surface Sz terminated by a critical line r ) rB(h) as shown in Figure 3a. The transition form the isotropic to the nematic phase at h ) 0 is first order, and the transition to the biaxial state at h < 0 and |h| small is also. If S were frozen, the transition to the biaxial state would be second order. As |h| increases, fluctuations in S decrease, and beyond a tricritical point tZ at (r, h, S) ) (rt, ht, St) the transition to the biaxial state becomes second order. To study the vicinity of this second-order transition and to locate the tricritical point, we integrate out fluctuations
S)-
|h|3
(36)
on this line, along which Dη > 0 for |h| > |ht|. At the tricritical point, Dη ) 0, and this condition along with eq 35 leads to
rt )
21 , 128
ht ) -
27 , 1024
St ) -
3 32
(37)
When |h| < |ht|, there is a first-order phase transition between the oblate uniaxial phase and the biaxial phase with a phase boundary, shown in Figure 3, that can be determined numerically. The biaxial region, SZ, in the phase diagram is shown in Figure 3. Since the energy of the biaxial phase is independent of the rotation angle θ (see below eq 19), a continuum of biaxial states coexist on SZ. We will refer to such surfaces as CC
3860 J. Phys. Chem. B, Vol. 113, No. 12, 2009
Ye and Lubensky
(continuous coexistence) surfaces and ones on which a discrete set of states coexist as DC (discrete coexistence) surfaces. Note that the biaxial region extends to a maximum value of r ) rm ) 3/8 at h ) hm ) -27/64, three times the value, rc ) 1 /8, of r at the mechanical critical point. As we shall see, it is this fact that is responsible for the existence of semisoft response well away from the nematic coexistence region. The Z-plane is divided into a region with uniaxial order (prolate for h > 0 and oblate for h < 0) in which S ) Su(r, h) and a region SZ with biaxial order (with h < 0). In the biaxial region SZ, the biaxial order parameter is nonzero and, from eqs 21b and 21c with σxx ) 0, satisfies
η2 ) -
r S S2 - 8 4 3
(38)
With this form of η, S from eq 21a with σxx ) 0 satisfies
8h + 3r + 6S + 32S2 ) 0
(39)
The solutions to these equations are
SB(r, h) ) ηB(r, h) )
(
3 1+ 32
1-
)
32 256 rh 3 9
[ 121 h - 323 (r + 2S (r, h))]
1/2
B
(40)
(41)
where we have chosen the positive solution for ηB(r, h). Note that SB(r, h) < 0; S does not undergo a change in sign at the transition to the biaxial phase. The full biaxial order parameter is (η1, η2) ) ηB(r,h)(cos 2θ, sin 2θ). It is a straightforward exercise to verify that these expressions reduce to S ) St ) -3/32 and η ) 0 at (r, h) ) (rt, ht) and that η ) 0 along the second-order line described by eq 35. More interesting is the limit h f 0-. In this case at r ) rN ) 1/12, η2 ) 1/64, and S ) -1/8. Thus, if we assume θ ) 0, we can obtain from eq 19 that uij ) 1/4(ninj - 1/3δij) with n along ex, which corresponds to nematic order at the clearing point (SN ) 1/4) with uniaxial anisotropy in the plane perpendicular to the z-axis. As discussed following eq 19, the ordering of the magnitudes of eigenvalues changes when ηB passes through |SB|/3. Equating ηB(r, h) and |SB(r, h)|/3 in eqs 40 and 41 yields the line
4 1 rex(h) ) - |h| + √6|h| 3 2
(42)
below which the direction associated with the largest magnitude eigenvalue is along e1 rather than along ez. In our convention, ez is still referred to as the direction of uniaxial anisotropy and e1 as the direction of biaxial anisotropy. Equations 40 and 41 also allow us to verify that the limit ηB(r, h) ) SB(r, h) at which u_ becomes uniaxial is never reached in SZ (on which h < 0). Thus, _u is biaxial throughout this region. In addition the energy is invariant with respect to arbitrary rotations of the biaxial axis e1 in the xy-plane, whether r < rex(h) or not. B. Full Phase Diagram in the r-h-σxx Space. Now we consider the full phase diagram31,36,37 of the minimal model defined by the Gibbs free energy of eq 18 in the space spanned by r, h, and σxx. This full phase diagram, which is shown in Figure 4, necessarily reflects the symmetries of g. The x-, y-,
Figure 4. (a) Phase diagram in the r-h-σxx space showing the SY and SZ (SX hidden) CC and the DX, DY, and DZ DC surfaces along with the tricritical points tX, tY, and tZ; (b) is the projection of panel a onto the h-σxx plane.
and z-directions are are all equivalent in fiso while fani and fext, respectively, pick out the z- and x-directions. In equilibrium (i.e., when _u obtains its value minimizing g) g satisfies the symmetry relation g(r,h,σxx) ) g(r,σxx,h). This follows because fiso is invariant under uxxTuzz. Thus the Z-plane with σxx ) 0 is symmetry equivalent to the X-plane with h ) 0, and turning on σxx near the Z-plane is equivalent to turning on h near the X-plane. The order parameter in the X-plane is given by eq 19 with ez and ex interchanged and with S ) Su(r,σxx) when the order parameter is uniaxial and S ) SB(r,σxx) and η ) ηB(r,σxx) when the order parameter is biaxial. There is an additional symmetry in g that follows from the traceless constraint on _u. Because uyy ) - uxx - uzz,
huzz + σxxuxx ) -huyy + (σxx - h)uxx
(43)
and because fiso is invariant under uyyTuzz,
g(r, h, σxx) ) g(r, -h, σxx - h)
(44)
This means that the Y-plane defined by σxx ) h is equivalent to the Z-plane with the opposite sign of h and that positive σxx h near the Y-plane is equivalent to positive σxx near the Z-plane. The order parameter in the Y-plane is given by eq 19 with ez and ey interchanged and with S ) Su(r, -h) when the order parameter is uniaxial and S ) SB(r, -h) and η ) ηB(r, -h) when the order parameter is biaxial. To reiterate, the phase structure of the Z-plane is replicated in the X-plane (h ) 0) and the Y-plane (σxx ) h) with respective preferred uniaxial order along ex and ey, critical points cX and cY, biaxial coexistence surfaces SX and SY, and tricritical points tX and tY. To fill in the 3D phase diagram, we need to consider perturbations away from the X-, Y-, and Z-planes. Continuity requires that the PN-N coexistence line be on a coexistence surface that exists at σxx * 0. Since increasing |σxx| from zero converts the uniaxial order parameter on the h > 0 Z-plane to a biaxial one, this must be a DC surface DZ on which two biaxial phases coexist. The directions of both uniaxial and biaxial order of the phases that coexist on this surface are not spontaneous: they have unique directions determined by the fields σxx and h. DZ is terminated by a critical line (r, h, σxx) ) (rZc (h), h, σZxx,c(h)) connecting the tricritical points tX and tY and passing through the mechanical critical point cZ. Turning on σxx near the SZ
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J. Phys. Chem. B, Vol. 113, No. 12, 2009 3861
surface favors alignment of the biaxial order along ex when σxx > 0 and along ey when σxx < 0. Thus σxx is an ordering field for biaxial order whereas a linear combination of h and σxx acts as a nonordering field.38,39 The topology of the phase diagram near tZ is that of the Blume-Emery-Griffiths model40 with DC surfaces DX and DY emerging from the first-order line NctZ terminating SZ. The DX and DY surfaces terminate, respectively, on the critical lines NctY and NctX in the Y- and X-planes. The DX and DY surfaces and their corresponding boundaries (r, h, σxx) ) (rXc (h), h, σXxx,c(h)) and (r, h, σxx) ) (rYc (h), h, σYxx,c(h)) can be obtained numerically or by symmetry transformation from DZ and rZc (h). The surfaces DX, DY, and DZ form an inverted cone with vertex at Nc. The SY-surface and its vicinity play a particularly important role in semisoft response, and it is useful to consider it in more detail. Recall that on this surface, σxx ) h, and the stress energy of eq 43 is equal to -huyy, which favors oblate uniaxial order along y (for h > 0). In the SY semisoft phase where there is nonzero biaxial order, uij takes the form of eq 19 with the cyclic replacement (x,y,z) f (z,x,y) so that e1 ) ((sin θ, 0, cos θ) and e2 ) (cos θ, 0, (sin θ), where θ is now the angle between the z-axis and the principal axis of biaxial order, and
u_ )
(
-
S - η1 3
0
η2
0
2S 3
0
η2
0
-
S + η1 3
)
(45)
Now the direction of uniaxial order, which like the uniaxial order in the Z-plane is oblate (S < 0) at high temperature, is along ey rather than along ez, and biaxial order is in the xzplane perpendicular to ey. At σxx ) h- (which corresponds to σxx ) 0- near the Z-plane) η2 ) 0 and η1 ) ηB(r, -h) > 0 implying that θ ) θ- ) 0 (recall, 0 < θ < π/2) and that the principal axis for biaxial order is along e1- ) (0,0,1) ) ez. At σxx ) h+, η2 ) 0 and η1 ) -ηB(r, -h) < 0, implying that θ ) θ+ ) π/2 and that the principal axis for biaxial order is along e1+ ) (ex. At σxx ) h(,
u( xx )
1 |S (r, -h)| ( ηB(r, h) 3 B
(46a)
( uzz )
1 |S (r, -h)| - ηB(r, h) 3 B
(46b)
where we used SB(r, -h) ) -|SB(r, -h)| < 0. Thus uxx and uzz undergo respective jumps of 2ηB(r, -h) and -2ηB(r, -h), and the principal biaxial axis rotates by π/2 from ez to (ex in crossing the SY surface. The boundary of the SY surface is r ) - urB(-h). At σxx ) h-, u_ ≡ u_-, and ηB(r, -h) ) (uzz xx)/2. C. Phase Diagram in Terms of r, h, and uxx. The phase diagram in the space defined by r, h, and uxx provides useful information, not readily apparent in the phase diagram in terms of r, h, and σxx, for the calculation of semisoft response. Rather than displaying a full 3D plot of the various regions of coexistence of different values of uxx, we will consider various cuts parallel to the r-σxx plane at different values of h as shown in Figure 5. These cuts, restricted to h > 0, show the two types of coexistence regions: the discrete coexistence (DC) regions associated with the surfaces DZ and DX and the continuous coexistence (CC) regions associated with the surface SY.
Figure 5. Phase diagrams in the r-uxx plane for h ) 0 (a), h ) 0.002 (b), h ) hc ) 1/192 (c), h ) 0.01 (d), h ) ht ) 27/1024 (e), and h ) 32/1024 (f), respectively. The dashed (red) lines represent initial equilibrium states at σxx ) 0. The dotted areas are discrete coexistence regions. The semisoft (or soft) regions are marked correspondingly.
The boundaries of the DC surfaces are determined by the strains uxx1 and uxx2 that coexist on them. Values of uxx between these two values are assumed to follow the lever rule like the density in a liquid-gas coexistence region: uxx ) λuxx1 + (1 λ)uxx2, where 0 e λ e 1. In a liquid-gas system or in a binary liquid in a closed container in the coexistence region, a single interface separates two regions of different density. Though the elastomer coexistence we consider here is formally equivalent to the liquid-gas or binary liquid problem in mean-field theory, boundary constraints will prevent the formation of a single boundary separating a region with uxx ) uxx1 from one with uxx ) uxx2, and in general a complicated polydomain structure will develop.20 Here, we ignore these structures. As we shall see, additional complications arise when the first PK stress σ1xx rather than the second PK stress σxx is held fixed. The strains uxx and uxz on the CC surfaces are determined by different rules. As discussed following eq 45, the energy is invariant with respect to rotations of the direction of biaxial anisotropy within the xz-plane so long as the magnitude of the biaxial order parameter remains fixed. On SY,
uxx )
1 |S (r, -h)| - ηB(r, -h) cos 2θ 3 B
(47)
uxz ) (ηB(r, -h) sin 2θ
(48)
1 uxx can take on any value between uxx ) /3|SB(r,-h)| - ηB(r, + 1 -h) and uxx ) /3|SB(r, -h)| + ηB(r, - h), and uxz can take on any value between -ηB(r, -h) and ηB(r, -h). If the strain uxx + is externally fixed to have any value between uxx and uxx, then uxz ) η2 (or equivalently θ) will adjust to ensure that η12 + η22 retains its equilibrium value of η2B. The process occurs at a particular point on SY where σxx ) h is fixed. Thus, as uxx is varied, σxx does not change, and the result is the semisoft
3862 J. Phys. Chem. B, Vol. 113, No. 12, 2009 stress-strain curve with a flat-plateau shown in Figure 1. For each value of uxx, there are two choices of uxz ) η2 ) ((ηB(r, -h)2 - η12)1/2. In an ideal system, the lowest energy state on the SY CC surface, for each value of uxx, is one in which uxz adopts one of these values uniformly throughout the sample. In real systems, boundary conditions will prevent the adoption of a spatially uniform value of uxz, and there will in general be complicated multidomain texture of both the positive and negative solution for uxz, predominately arranged in a striped array in the xz-plane.13,19,41 Figure 5 shows both the DC and CC coexistence regions. It also shows the equilibrium curve for uxx as a function of r at σxx ) 0 for each value of h. Note that uxx is negative along this curve because when σxx ) 0 the system is prolate uniaxial along the z-axis. The full coexistence regions are accessed by changing σxx. Consider first h < ht, and define rm(h) ) min[rZc (h), rXc (h)], rM(h) ) max[rZc (h), rXc (h)]. For r > rM(h), there is no coexistence region; for rm(h) < r < rM(h), there is one discrete coexistence region from one of the DC-surfaces; when rB(-h) < r < rM(h), there are two discrete coexistence regions; and when r < rB(-h), there is the semisoft region arising from the SY surface. When |h| g |ht|, the DC-surfaces can no longer be accessed, and there is only the semisoft region. For any given r, the values uxx and u+ xx at the lower and upper boundaries of the semisoft region are given by eq 46a. The boundaries of discrete coexistence regions can be obtained numerically from the shape of f ) fiso + fani as a function of uxx since in this case uxz ) 0 and uzz can be relaxed. D. Semisoft Response. Having determined the global phase diagram of the minimal model, our next task is to determine the strain uxx in response to stress σxx and to investigate the origin and domain of existence of semisoft response. In particular, starting at some point PZ ) (r,h,0) in the σxx ) 0 0 ) 2Su(r, h)/3 ) -2u0xx, we Z-plane with equilibrium strains uzz wish to determine how δuxx ) uxx - u0xx behaves as a function of σxx and how the semisoft plateau arises. It is clear from our discussion of the phase diagram in the preceding subsection that upon increasing σxx from the zero on the Z-plane, either the DC surface DX, DZ and then DX, the CC surface SY, or no coexistence surface at all is crossed. If no surface is crossed, the uxx-σxx curve will be smooth and monotonic. Crossing either the DX, the DZ, or the SY surface will introduce structure into the uxx-σxx curve but of a different nature. We define semisoft response1 as one in which there is a plateau in the stress-strain curve produced by the continuous development of shear strain uxz, which only arises as the surface SY is crossed on increasing σxx. Though in real systems, domains of opposite sign of uxz will always develop as SY is crossed, there is an ideal theoretical limit in which only a monodomain with one sign of uxz forms. This is in contrast to crossing a DC surface when there is at least one domain boundary. In this section, we will focus mostly on true semisoft behavior in which the SY surface is traversed. The first question then is for what starting values of r and h on the h > 0, Z-plane is semisoft response possible. These are clearly the values that span the SY surface on the Y-plane, that is, there will be semisoft response for r < rB(-h). Figure 2a shows the r,h > 0 part of the Z-plane with the PN-N coexistence line terminating at the critical point cZ and the SY, DX, and DZ surfaces projected on to it. If the point PZ ) (r,h,0) lies within the projected region of the SY-surface, there will be semisoft response upon increasing σxx. If PZ lies within the projected region of the DX or DZ surfaces, there will be Martensitic-like polydomain response20 upon increasing σxx. Otherwise, uxx will
Ye and Lubensky depend smoothly on σxx. This figure shows that for h < hc, rB(h) lies below the PN-N coexistence line and that for starting points PZ in the vicinity of the critical point cZ, there will be Martensitic rather than semisoft behavior. In addition, for a given h semisoft response will always occur for r sufficiently small (including negative). As we have emphasized, semisoft behavior is a consequence of the existence of biaxial order on SY that spontaneously breaks the rotational symmetry about the y-axis of the high-temperature oblate uniaxial phase. For comparison with soft behavior when h ) 035 and with semisoft behavior calculated in refs 1 and 14, it is instructive to express the strain on the semisoft plateau relative to the reference state at the σxx ) h- side of the plateau _ -TΛ _- - δ _)/2. Because uyy where Λ _ )Λ _ - and _u ) u_- ) (Λ remains constant on this plateau, we focus on the xz-subspace, and define
∆u_P ) _uP - _u-
(
) ηB(r, -h)
1 - cos 2θ sin 2θ sin 2θ -(1 - cos 2θ)
)
≡Λ _ -Tu_PΛ _-
(49)
where _u p is the strain _u on the plateau, and _u′P is the strain on the plateau relative to the reference system at σxx ) h-, which can be simplified to
(
1 ∆u_P ) (F - 1) 4
1 - cos 2θ
1 sin 2θ √F
1 sin 2θ √F
1 - (1 - cos 2θ) F
)
(50)
2 where F ) (Λzz /Λxx) . This is exactly the form of the strain in the regime of soft response in soft rather than semisoft systems.35 In the soft case, the reference state is the uniaxial nematic state, with anisotropy axis taken for convenience along z, and the strain is soft in any plane containing the z axis. In the semisoft case, the reference state at σxx ) h- is biaxial, and the system only exhibits semisoft response in the xz plane, which is perpendicular to the oblate uniaxial axis along y and which contains the biaxial axis. We show in appendix B that the same strain calculated within the generalized neoclassical model1,14 also has the form of eq 50 and in addition that full strain in this model measured relative to an isotropic reference state is biaxial with a form identical to that of eq 45. This provides strong support for the hypothesis that semisoft response is always associated with the development of biaxial order such as that encountered on SY. We also display in appendix B the full strain tensor for the soft case to show that is the appropriately defined uniaxial limit of the biaxial semisoft case. Our global phase diagram also provides for us the form of the stress-strain response for h > 0 and σxx < 0 and for h < 0 and σxx of either sign. The former case is trivial. There are no CC phase boundaries in the octant h > 0, σxx < 0 in the global phase diagram of Figure 4. Thus if uxx is decreased from its equilibrium value on the positive Z-plane, σxx will smoothly decrease from zero or exhibit Martensitic response. The latter case is more complicated. If r > rB(h), the order parameter is oblate uniaxial at σxx ) 0. In this case, either no phase boundaries are encountered as |σxx| is increased from zero, and σxx will be a monotonic increasing function of uxx; or a DC phase boundary will be encountered, and there will be martensitic
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response. On the other hand, for r < rB(h) and σxx ) 0, the system is in a biaxial state with the biaxial axis pointing along 1 some vector e1 in the xy-plane and uxx between uxx ) /3|SB| 1 + ηB and u+ xx ) /3|SB| + ηB. If uxx < uxx < uxx, changing uxx will rotate e1 at zero stress. In other words, the system will exhibit soft and not semisoft response. It is unlikely that this ideal soft situation will be realized in real systems because it is virtually impossible to create a monodomain biaxial state without freezing in an anisotropy direction perpendicular to the already established uniaxial direction. If this direction is frozen in, the response will be semisoft rather than soft. E. Ward Identities and Stress-Strain Relations. General properties of the δuxx-σxx stress-strain curve follow from a Ward identity.11 The isotropic energy, fiso(u_), is invariant under _ is arbitrary rotations of _u, that is, under _u fU _ u_U _ -1 where U __u, where hij ) heziezj, any rotation matrix. Thus for fani ) -Trh
f(U _ u_U _ -1) ) fiso(u_) - Trh _U _ u_U _ -1
(51)
for any U _ and, in particular, for one describing an infinitesimal rotation by θ about the y-axis with components Uij ) δij + εyijθ, where εijk is the Levi-Civita antisymmetric tensor. Differentiating eq 51 with respect to θ using df/dθ ) (∂f/∂uij)(duij/dθ), _U _ u_U _ -1)/dθ ) duij/dθ ) εyikukj - uikεykj, dfiso/dθ ) 0, and d(Trh 2huxz, we obtain the Ward identity
2σxz(uzz - uxx) ) 2(σzz + h - σxx)uxz
(52)
where, as before, σij is the second PK stress tensor, ∂f/∂uij. This identity can be derived either by treating uij as an arbitrary tensor and then applying the symmetry condition on uij and σij after the linear term in θ has been extracted; or it can be derived by using the Voigt notation in which uij is represented by its six independent components, in which case the xz component of σijuij is 2σxzuxz, and ∂f/∂uxz ) 2σxz. Equation (52) applies for any fiso, including ones with no compressibility constraint, so long as fani is linear in _u. In the semisoft geometry, σxz ) σzz ) 0, but σxx > 0, and (h - σxx)uxz ) 0. Thus, either uxz ) 0 or h ) σxx: states with nonzero shear strain uxz can only exist on the Y-plane, and, then, only on its SY surface. As σxx is increased from zero for PZ in the projection of SY, δuxx and δuzz will deviate from zero but uxz will remain zero until SY is reached. At this point uxz becomes nonzero, and as we shall see in more detail below, the δuxx-σxx curve becomes flat. Once σxx exceeds h, uxz must again be zero, and a further increase of σxx leads to a further increase of δuxx. The Ward identity provides us with an expression for the elastic modulus C5 associated with uxz in states with uxz ) 0 (A more detailed analysis of elastic response and Ward identities will appear in a separate publication42). Using the Voigt notation,
C5 )
∂2f ∂u2xz
|
) uxz)0
2σxz uxz
|
uxz)0
2(h - σxx) ) uzz - uxx
(53)
Thus on the Z-plane, σxx ) 0, where uzz ) 2/3Su(r, h) ) -2uxx,
C5 )
2h 8 ) 2 r - Su(r, h) + S2u(r, h) S 3
[
]
(54)
As σxx f h(, uzz f |(SB(r, -h)/3)|-ηB(r, -h), uxx f |(SB(r, -h)/3)|(ηB(r, -h), and uzz - uxx f -2ηB(r, -h) and
C5 f
|h - σxx | ηB(r, -h)
(55)
This establishes that the modulus C5 for δuxz vanishes at the two extremes of the semisoft plateau where σxx ) h(. We can now fill in some quantitative details about the stress-strain curve. We begin by considering behavior for small σxx near the Z-plane. Response in this region is determined by the elastic energy f expanded to harmonic order in the deviation, δuij, of the strain from its equilibrium value on the Z-plane. We are interested in the response of uxx to σxx. At any point P ) (r, h, σxx), the Gibbs free energy can be expanded in powers of the deviation δuij(P) of the strain from its equilibrium value uij(P) at that point. If σxx differs by δσxx from its value at P, there will be a term -δσxxδuxx in δg ) g - g(P). The nextorder contributions to δg are harmonic in δuij(P). Strains δuxy or δuyz are not induced by σxx, and because of the constraint δuyy ) -δuxx - δuzz, we can express the free energy in terms of δuxx, δuzz, and δuxz only:
1 1 2 δg ) Cxx(P)δu2xx + Czz(P)δuzz + Cxz(P)δuxxδuzz + 2 2 1 C (P)u2xz - δσxxδuxx (56) 2 5 where it is understood that δuij ) δuij(P). General expressions for the elastic constants in the expression are given in appendix C. An in-depth treatment of elastic response and tensors including a discussion of the relation between the elastic tensors of the traceless model and the complete physical tensor (with compressible components) will be presented in reference 42. To determine the response of δuxx to the stress σxx, we can integrate over δuzz to obtain
1 1 δg ) Cxx(P)δu2xx + C5(P)u2xz - δσxxδuxx 2 2
(57)
where Cxx ′ ) Cxx(P) - [C2xz(P)/Czz(P)]. Then δuxx(P) ) δσxx/ ′ (P). Of particular interest is the response on the Z plane where Cxx ′ (PZ) and P ) PZ ) (r, h, 0) for which δuxx(PZ) ≡ δuxx ) σxx/Cxx on the Y-plane where P ) PY( ) (r, h, h ( ) and δuxx(PY() ) (σxx - h()/Cxx ′ (PY). Expressions for Cxx ′ (PY) and other elastic moduli are given in appendix C. We can now provide an essentially complete description of the δuxx-σxx stress-strain curve. For small σxx, δuxx grows ′ (PZ) > 0; it continues to linearly from zero with slope 1/Cxx increase with σxx until σxx ) h- at which point δuxx ) uxx(σxx ) h) - u0xx ) 1/3[|SB(r, -h)| + Su(r, h)] - ηB(r, -h). The strain δuxx increases further at constant σxx ) h by rotating the biaxial order parameter while keeping its magnitude fixed, that is, by developing nonzero shear strain uxz ) η2 such that η12 ) η2B(r, -h) - η22:
1 δuxx ) [|SB(r, -h)| + Su(r, h)] + ηB(r, -h) cos 2θ 3
(58)
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where θ runs from π/2 at σxx ) h- to 0 at σxx ) h+. For σxx > h, δuxx increases monotonically with σxx with initial slope ′ (PY+). 1/Cxx F. First Piola-Kirchhoff Stress. The second PK stress tensor σij, being the derivative of the free energy with respect to the strain uij, has the virtue of relative simplicity: it depends only on uij and not independently on the deformation tensor Λij. As we have already noted, however, the second PK stress tensor is not a physical stress tensor whose magnitude can be controlled by experimentally fixing an external force. In stretching experiments such as those that probe semisoft behavior, the external stretching force is controlled. Since the area in the reference space of the surfaces across which the external forces are applied does not change, these experiments control the first PK stress tensor σIij (and not the Cauchy stress tensor, whose ratio to the external force changes as the area of surfaces in the target space change). We, therefore, consider in this subsection the phase diagrams and stress-strain curves when σIxx and not σxx is controlled and its conjugate field Λxx rather than uxx is measured. Before studying phase diagrams when σIxx or Λxx are varied, we make some simple observations that explain most of the differences between these phase diagrams and those arising when σxx and uxx are varied. The free energy density f(u_, r, h) remains the same whether σxx or σIxx is varied. It depends only on r, h, and u_, which adopts a spatially uniform equilibrium value in the phases we consider. Thus, each equilibrium phase has a well-defined spatially uniform second PK stress tensor σij ) ∂f/∂uij. In particular, for stresses applied along ex, the only nonvanishing component of σij is σxx, and from eq 6,
σIxx ) Λxxσxx
(59)
This relation minimizes a modified Gibbs free energy density
gI ) f(u_) - σIxxΛxx
(60)
Because _u ) 1/2(Λ _ TΛ _ -δ _), gI is a function of Λij. At the minima of gI, Λxx will reach the maximum value that can be consistent with the values of uij for any given σIxx. Because uxx ) 1/2(Λ2xx + Λ2zx - 1), we thus expect Λzx ) 0, and indeed we can verify this result by considering the full equations of state for all Λij. With Λzx ) 0, there is a one-to-one correspondence between uxx and Λxx:
Λxx ) √1 + 2uxx
or
1 uxx ) (Λ2xx - 1) 2
(61)
2 In addition, uxz ) uzx ) 1/2ΛxxΛxz and uzz ) 1/2(Λzz + Λ2xz - 1). I I Thus we can take g ) g (Λxx, uzz, uxz) to be a function of Λxx, uzz, and uxz rather than Λxx, Λxz, and Λzz. We now return to the implication of eq 59. In the semisoft + plateau, σxx is a constant for uxx < uxx < uxx or equivalently for + ( ( 1/2 Λxx < Λxx < Λxx, where Λxx ) (1 + 2uxx) , implying that σIxx rises linearly with Λxx on the plateau as shown in Figure 1c and Figure 6b. Thus, whereas there are many values of uxx corresponding to the single plateau value of σxx, there is a unique value of σIxx for each value of Λxx. As we have seen, for each value of uxx in the semisoft regime, there are two uniquely determined values of uxz of equal magnitude and opposite sign, and, consequently, for each value of σIxx, there is a unique value of Λxx and an associated pair of values of Λxz ) (2Λ-1 xx |uxz|.
Figure 6. (a) Soft (full line) and semisoft (dashed and dotted line) stress-strain curves at r˜ ) 0.08 with h˜ ) 0, h˜ ) 0.8h˜c, and h˜ ) 4h˜c, respectively. (b) Semisoft curve of σIxx as a function of Λxx at r ) 0.08w and h ) 2hc, where we have set V ) w and σIxx is plotted in units of w.
Figure 7. (a-c) Phase diagrams in the r-σIxx plane for h ) 0.018w, h ) 0.013w, and h ) 0.002w, respectively. r and σIxx are measured in units of w, and, to simplify calculations, V was set equal to w. Dashed lines represent second-order phase boundaries and solid lines first-order phase boundaries. Points CX and CZ are critical points associated, respectively, with the termination of the DX and DZ surfaces. TX and TZ are tricritcal points separating first-order from second-order phase boundaries and CEX and CEZ are critical end points at which secondorder phase boundaries of the semisoft region terminate on the firstorder boundaries. Note that as h becomes smaller, the width of the semisoft region diminishes. (d) Phase diagram for second PK stress in the r-σxx plane for h ) 0.002w for comparison. The two slightly separated second-order boundaries in panel c are replaced by a single CC line in panel d.
This implies that the 2D SY surface in the r-h-σxx space opens into an extended 3D semisoft domain in the r-h-σIxx space. The DC surfaces Dx and Dz of Figure 4 continue to exist in the r-h-σIxx space; they connect to the semisoft domain in a manner that we will discuss further below. I , we To determine the phase diagram for external σxx I _ does minimized g (Λxx, uzz, uxz) numerically. Since, unlike _u, Λ not have a useful reduced form, we need to return to the nonreduced form in this subsection. For these calculations, we let the quantities without a “∼” stand for their nonreduced form and those with the “∼” represent the reduced form. Phase diagrams in r-σIxx planes of constant h are shown in Figure 7, and a phase diagram in the r-Λxx plane is shown in Figure 8. To visualize the nature of the 3D extended semisoft domain and its boundaries, it is instructive to consider r-σxx and r-σIxx planes of constant h, respectively, in the r-h-σxx and r-h-σIxx spaces. We calculated phase diagrams, shown in Figure 7, in the r-σIxx plane by minimizing gI(Λxx, uzz, uxz) numerically. For second PK stress, when h > |ht|, the intersection of the SY surface with an r-σxx plane yields a CC line parallel to the r-axis at σxx ) h and terminated by a critical point at r ) rB(|h|). For h < |ht|, the CC line terminates at a triple point from which emerge two DC coexistence lines, arising from intersection with the DZ and DX surfaces, that terminate at critical points as shown
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Λ′xx )
Λxx Λ0xx
)
√1 + 2uxx
√1 + 2u0xx
(64)
and the first PK stress σ′Ixx is related to σIxx via
Figure 8. Phase diagrams in the r-Λxx plane for h ) 0.002w (corresponding to Figure 7c) showing the semisoft region and regions of discrete coexistence (vertical dotted lines). Panel b is a blow-up of panel a. The dashed line (red online) at the bottom of these figures represents the equilibrium states with no stress.
in Figure 7d. For first PK stress, the tricritical point where the second-order transition first turns to a first-order transition remains at r ) rt and h ) ht as it does for second PK stress. For h > |ht| the intersection of the extended semisoft domain with an r-σIxx plane is an extended 2D domain terminated by a second-order phase boundary shaped like a parabola with axis parallel to r and opening toward negative r. For h < |ht|, this boundary develops a first-order part at its apex that terminates at two tricritical points as shown in Figure 7a. At smaller h, the effects of the DZ and then DX surfaces appear in the form of first order lines connected to the first-order boundary of the semisoft domain but extending beyond it and terminating at critical points as shown in Figure 7b,c. The remaining secondorder boundaries of the semisoft domain terminate on this firstorder boundary. Figure 8 shows the phase diagram in r-Λxx for the same value of h as that of the r-σIxx phase diagram of Figure 7c. It shows the semisoft region with continuously varying value of Λxz (and Λxx) and regions of coexistence of different values of Λxx. G. Stress and Strain Relative to the Initial State. To obtain a consistent description of global phase diagrams, we have measured deformations Λ _ and strains _u relative to the hightemperature isotropic state (we ignore thermal expansion). However, as discussed in section IIC, in real experiments, deformations Λ _ ′ and strains _u′ are measured relative to the equilibrium state at a given temperature. In fact, it is often not possible to access the isotropic state and to establish a reference to it. Therefore, to make contact with real experiments, we need to transform from Λ _ to Λ _ ′ and from _u to _u′. This is easily accomplished using the development of section IIC. In our problem, equilibrium reference states are characterized by a diagonal deformation tensor Λ _ 0 with nonzero components 0 and associated strain tensor _u0 ) 1/2(Λ _ TΛ _Λ0xx, Λ0yy, and Λzz δ _). From eq 12,
(Λ0xx)-2δuxx
u xx )
(62)
and
0 -1
σ xx ) (det Λ _ )
(Λ0xx)2σxx
)
Λ0xx 0 Λ0yyΛzz
σxx
(63)
Thus at any given temperature, the u′xx-σ′xx curve is a simple rescaling of the δuxx-σxx curve. The flat plateau in the δuxx-σxx curve remains a flat plateau in the u′xx-σ′xx curve. Similar considerations apply to the curves of first PK stress versus deformation. The deformation Λ′xx is related to Λxx via
σ′Ixx ) (det Λ _ 0)-1Λ0xxσIxx ) (det Λ _ 0)-1Λ02 xx Λ′xxσxx ) Λ′xxσ′xx (65) The transformation from the unprimed variables to the primed ones involves multiplication by constant factors depending only on Λ _ 0, and the σ′Ixx-Λ′xx curve is simply a rescaling of the I σxx-Λxx curve. In particular, the first PK stress on the semisoft plateau grows linearly with Λxx and Λ′xx in the two cases, respectively. Thus, using eq 64 and eq 65, we can then easily obtain the σ′Ixx-Λ′xx stress-strain curve, which is shown in Figure 6b, from the σxx-uxx curve. These two curves are similar, but the flat plateau in the σ′Ixx-Λ′xx curve rises linearly with Λ′xx, and there is a unique value of Λ′xx for each value of σ′Ixx. IV. Other Models We can now consider modifications of the minimal model. A simple modification is to replace the constraint Tru _ ) 0 with the real volume constraint det Λ _ ) 1, which will be discussed in section IVA. This replacement does not change the validity of the Ward identity (eq 52) and the resulting phase diagram has the same structure as that for Tru _ ) 0 but with different boundaries for the CC and DC surfaces. Section IVB investigates other modifications of the minimal model which replace fani with nonlinear functions of uzz. Modifications of this kind can spread the CC surface SY into a finite volume or convert it to a DC surface. A. Real Incompressible Constraint. For large deformations, the volume change of a body is proportional to det Λ _ . The incompressible constraint is applied by requiring det Λ _ ) 1 or _ - 1)2 to the free by adding a compression term 1/2B(det Λ energy with the compression modulus B infinitely large. Here we will use the latter approach. Furthermore, we will change _Λ _ T - 1)2 ) 1/2B[det (1 + the compression term to 1/2B(det Λ 2 2u _) - 1] to simplify calculation. The Gibbs energy density is thus,
1 g ) B[det(1 + 2u _) - 1]2 + fiso(u_t) - huzz - σxxuxx (66) 2 where _ut ) u_ - 1/3(Tru _)δ _ is the traceless part of _u. In this case, the Ward identity (eq 52) still holds, which means the semisoftness still stays on the σxx ) h plane (or SY surface). However, although there is still symmetry between Sz and Sx, the SY surface is now different from Sz and Sx since _u is not traceless anymore. In the following we will determine the phase boundary of the SY surface on the h ) σxx plane, and the boundaries for the DC surfaces and for Sx and Sz will be ignored since we are only concerned with the semisoft behavior. To determine the boundary of SY, we follow a similar procedure to that used in the minimal model. As shown in appendix D, we can still determine the tricritical point and the second-order part of the phase boundary analytically, while the first-order part of the phase boundary can only be determined numerically. The projection of the boundary of SY onto the r-h plane is shown in Figure 9. We can see the phase diagram now
3866 J. Phys. Chem. B, Vol. 113, No. 12, 2009
Figure 9. (Color online) Projection of the boundary of SY onto the r-h plane for B ) 100w and V ) w. (b) Region inside the small (green) rectangle in panel a. The unit of r and h is w. The thick lines represent first-order phase boundaries, the thin lines represent second-order phase boundaries, and the dots are tricritical points. The dashed line in panel b is the paranematic-nematic coexistence line. For comparison, we also show the corresponding curves obtained by using Tru_ ) 0, which are represented as dotted (red) lines. The paranematic-nematic coexistence line for Tru _ ) 0 is right beneath the dashed line in panel b.
has structure similar to that of the minimal model, but with different boundaries. For B ) 100w and V ) w, the mechanical critical point is at (rc, hc) ) (0.1279w, 0.0052w), which is determined numerically, and the tricritical point is at (rt, ht) ) (0.1900w, 0.0247w). B. Nonlinear Symmetry Breaking. Now we consider what happens if fani is a nonlinear function of uzz. We will still assume Tru_ ) 0 and use the second Piola-Kirchhoff stress for calculation convenience since either replacing Tru _ ) 0 with det Λ _ ) 1 or changing σxx to σIxx does not change the result qualitatively. In addition, we can use the reduced form again given the fact that there are no components of Λ _ involved formally in this case. However, the Ward identity (eq 52) does not hold for a nonlinear symmetry breaking term. Thus, we can only solve the problem numerically rather than analytically. The stress-strain curve and the phase diagram can be obtained by numerically minimizing the reduced Gibbs energy density, g˜ ) ˜fiso + ˜fani - σ˜ xxu˜xx, over u_˜ . Figures in the left column of Figure 10 show stress-strain curves for different ˜fani values, in which linear, quadratic, and cubic terms of u˜zz are included. In Figure 10 panels a and e, stress resulting from homogeneous deformation does not rise monotonically with strain, and therefore we expect coexistence of two different states. The corresponding phase diagrams on the r˜-σ˜ xx plane for these ˜fani values are shown schematically in the right column of Figure 10. We can see the CC surface SY can either be converted to a DC surface or spread into a finite volume. When there is no linear term of uzz in fani, the CC surface SY in the minimal model will be converted to a DC surface. If fani 2 ) -huzz , two states coexist,37 whereas with other forms such as might arise in an hexagonal lattice, three or more discrete states might coexist.37 In this case, rather than exhibiting a homogeneous rotation of the biaxial order parameter (if boundary conditions are ignored) in response to an imposed uxx, samples will break up into discrete domains of the allowed states. In other words, their response to external stress will be martensitic20 rather than semisoft.
Ye and Lubensky
Figure 10. Stress-strain curves and schematic phase diagrams on the r˜-σ˜ xx plane. The stress-strain curves at r˜ ) 0.08 for ˜fani ) -0.01u˜zz 2 ˜ 2 2 - 0.005u˜zz , fani ) -0.01u˜zz + 0.005u˜zz , and ˜fani ) -0.01u˜zz + 0.005u˜zz 3 - 0.02u˜zz are shown, respectively, in panels a, c, and e, where solid curves represent stress-strain curves associated with homogeneous deformation and dashed lines represent stress associated with coexistence of states. The inserted figure in panel e is the first half of the stress plateau shown in another scale, which shows the shape of the plateau more clearly. The three figures in the right column show schematic phase diagrams on the r˜-σ˜ xx plane conjectured from the corresponding stress-strain curve in the left column. The points Tp, C, and Cp in panels b, d, and f are, respectively, triple points, liquid-gas-like critical points, and critical end points. The b diagram is similar to those for the minimal model, where all transitions are first order, but the first-order CC line from SY is converted to a first-order DC line; in panels d and f the first-order line from SY is replaced by a surface terminated by two second-order (dashed) lines or one firstorder DC line and one second-order line.
V. Neoclassical Model The generalized neoclassical model1,14,32 can also be discussed in our language. In this model, the shear moduli have entropic origin, and the semisoftness is caused by the compositional fluctuation of individual polymer chains comprising the crosslinked network. The elastic free energy of the polymer network is an average over individual chains,
1 fnet ) µ〈Tr(Λ _ T_l -1) - ln det(_l 0_l -1)〉 _ _l 0Λ 2
(67)
where Λ _ is the deformation tensor relative to the state at the time of cross-linking and µ is the shear modulus. Here we have µ ) nskBT, where ns is the volume density of chain segments and T is temperature; _l0 and _l are the step-length tensors of the state at the time of cross-linking and of the measurement state, respectively. The existence of nematic order leads to the step-length tensors being anisotropic, and we can assume
_l 0 ) l0(δ _ + RQ0)
(68a)
_l ) l0(δ _ + RQ)
(68b)
where l0 is a length and R is the coupling parameter varying with polymer chains. In this case the average in eq 67 can be taken over R, whose average and variance are, respectively, 〈R〉 and 〈δR2〉.
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As phase transitions are involved in our investigation, we need the energy contribution from the liquid crystal part, which will be chosen to have the usual Landau-de-Gennes form,
1 fQ ) rQTrQ2 - w3TrQ3 + w4(TrQ2)2 2
(69)
Terms in gradients of Q have been ignored since we are only concerned with homogeneous deformations. The coefficient r in the effective _u-only theory decreases and becomes negative as the rQ in fQ decreases, and thus the phase transition occurs. Pulling along a direction perpendicular to the initial nematic director n0 causes not only rotation of the nematic director n but also a biaxial order since now the symmetry in the plane perpendicular to n0 is broken. Therefore, we cannot simply _ - nn). Instead, we use the assume _l ) l|nn + l⊥ (δ approximation that -1 _l -1 ≈ l-1 _ - δRg-1Q + δR2(g-1Q)2] 0 g [δ
Figure 11. Coexistence lines and mechanical critical points for a system subjected to no external stress: (a) on the r-Q0 plane at (〈δR2〉)1/2 ) 0.2; (b) on the r-〈δR2〉 plane at Q0 ) 0.2. The unit of r is w3.
(70)
where δR ) R - 〈R〉 and g_ ) δ _ + 〈R〉Q _ . We then expand fnet over δR. To the linear order of 〈δR2〉, the first term in eq 67 is
[
{
1 〈δR2〉 -1 f1 ) µ TrΛ _ g0Λ _ Tg-1 δ _g Q+ 2 〈R〉 〈δR2〉 〈δR2〉(g-1Q)2 + TrΛ _Λ _ Tg-2Q (71) 〈R〉
]
}
_ + 〈R〉 Q0. We have no compact expression for where g0 ) δ the expansion from the second term in eq 67 since the determinant of a matrix exists. Without Λ _ present, this part actually does not contribute to the elastic energy; it merely modifies fQ. _ rg0-1/2, we obtain that in terms of the Setting Λ _ ) (det g0)1/6Λ new deformation tensor Λ _r
{
[
1 〈δR2〉 -1 f1 ) µ(det g0)1/3 TrΛ _ rΛ _ rTg-1 δ _g Q+ 2 〈R〉 〈δR2〉 〈δR2〉(g-1Q)2 + _ rTg-2Q (72) TrΛ _ rg0-1Λ 〈R〉
]
}
f1 now consists of an isotropic part, the first term in the above expression, which is invariant under simultaneous rotations of Λ _ r and Q in the target space and under rotations of Λ _ r in the corresponding reference space, and a semisoft anisotropic energy,14 the second term in the above expression, that breaks rotational symmetry in the reference space. The semisoft term vanishes if 〈δR2〉f0, or it becomes an isotropic term if the system is cross-linked in an isotropic state (g0 ) δ _). Namely, the semisoft behavior is possible only when both of the following conditions satisfied: the compositional fluctuation exists ((〈δR2〉 * 0)) and the system is cross-linked in a nematic _). state (g0 * δ Phase diagrams of this model can be obtained by numerically minimizing over components of Λ _ r and Q the Gibbs energy
g ) fnet(〈R〉, 〈δR2〉, Q0) + fQ - σIxxΛrxx
(73)
where Q0 is the order parameter of the initial nematic state and n0 is chosen to be aligned along the z-axis. The incompressibility
Figure 12. Phase diagrams on the r-σIxx plane for various Q0 and 〈δR2〉. The dashed lines represent boundaries of the semisoft region, the solid lines are coexistence lines of two states, and the black dots represent critical points. The figures in the left column show how phase diagrams change with Q0 for (〈δR2〉)1/2 ) 0.2: (a) Q0 ) 0.01, (c) Q0 ) 0.02, and (e) Q0 ) 0.04. The figures in the right column show how phase diagrams change with 〈δR2〉 for Q0 ) 0.2: (b) (〈δR2〉)1/2 ) 0.2, (d) (〈δR2〉)1/2 ) 0.4, and (f) (〈δR2〉)1/2 ) 0.5. Graphs b and c are the same figures shown in different scales. The unit of r and σIxx is w3.
constraint det Λ _ ) 1 is applied during the numerical minimization. Since we are mainly interested in effects of changing Q0 and 〈δR2〉, we fix the value of 〈R〉 and assume 〈R〉 ) 2, which is the value of the coupling constant for a freely jointed nematic main chain polymer.1 In addition, we set µ ) w4 ) w3 for calculation convenience. The real values of these parameters may be quite different from the values we have chosen, but this difference should not qualitatively change our results. Figure 11 shows the phase diagram in the r-Q0 plane for (〈δR2〉)1/2 ) 0.2 and the diagram on the r-〈δR2〉 plane for Q0 ) 0.2 in the case of no external stress. For Q0 ) 0.2, the mechanical critical point is approximately located at (r ) 0.112w3, (〈δR2〉)1/2 ) 0.189); and for (〈δR2〉)1/2 ) 0.2, we have rc ≈ 0.104w3 and Q0c ≈ 0.179. The phase diagrams on the r - σIxx plane for various Q0 and 〈δR2〉 are shown in Figure 12, from which we can see that in this model semisoft behavior also persists well above the mechanical critical point. Stress-strain curves resulting from this model are shown as solid lines in Figure 13, where stress and strain are defined over the equilibrium state of no external stress. For comparison, we also show stress-strain curves resulting from a uniaxial Q, which are shown as dashed and dotted lines in Figure 13. Dashed lines are obtained when the magnitude of Q is allowed to change, and dotted lines are obtained when changes in the
3868 J. Phys. Chem. B, Vol. 113, No. 12, 2009
Figure 13. Stress-strain curves: (a) at Q0 ) 0.2, (〈δR2〉)1/2 ) 0.2, and r ) 0.09w3; (b) at Q0 ) 0.4, (〈δR2〉)1/2 ) 0.2, and r ) 0.19w3; (c) Q0 ) 0.2, (〈δR2〉)1/2 ) 0.4, and r ) 0.09w3; (d) Q0 ) 0.2, (〈δR2〉)1/2 ) 0.4, and r ) 0. Λ′xx and σ′Ixx are defined relative to the equilibrium state of no stress, and the unit of σ′Ixx is w3. To obtain solid lines, we allow Q _ to take its optimal value, as we have done in calculating phase diagrams where both the magnitude of Q _ and the biaxial order parameter are allowed to change. The dashed and dotted lines were calculated with Q _ restricted to be uniaxial. The magnitude of Q _ was free to vary in the calculation of the dashed line but was kept constant (i.e., Q _ was only allowed to rotate) in the calculation of the dotted lines. The insert in panel c shows the curves of that panel with an expanded vertical scale.
magnitude of Q is suppressed.1 The rate of increase of stress with strain beyond the semisoft plateau is considerably larger when the magnitude of Q is fixed than when it is allowed to relax. The introduction of the biaxial order has a large effect on the stress-strain curve near the phase boundary for large 〈δR2〉 or large Q0, shortening the length and lowering the height of the plateau, which is shown clearly in the inserted figure in Figure 13c. However, the biaxial order has little effect on the stress-strain curve for small 〈δR2〉 and Q0 (Figure 13a) or for samples deep in the nematic phase (Figure 13d). VI. Summary Nematic elastomers exhibit complex nonlinear elastic response. Of particular interest is the semisoft response that occurs in systems with a frozen-in weak uniaxial anisotropy. This response is characterized by nearly flat plateau in the curve of stress (σxx or σIxx) perpendicular to the initial anisotropy direction along z versus strain (uxx or Λxx) in the same direction. The
Ye and Lubensky plateau is a manifestation of a stress-induced phase transition to a state with off-diagonal strain (uxz of Λxz). Thus, phase diagrams in the space defined by temperature (r), anisotropy field (h), and external stress (σxx or σIxx) contain full information about when semisoft response will be encountered. In this paper, we derived phase diagrams and stress-strain response for a number of models of semisoft elastomers. A general result that emerges for all of these models is that properties at zero external stress provide no information about semisoft behavior, which is a fundamentally nonlinear phenomenon. Thus, in particular, measurements of linearized elastic response at zero stress provide no indication of whether or not a material will exhibit semisoft response. Our calculations also show that under appropriate conditions, Martensitic rather than semisoft response may occur. Another result that emerges from our study of simply models is that semisoft response is associated with the existence at nonzero σxx of a biaxial phase phase that spontaneously breaks the rotational symmetry of a high-temperature oblate unixial phase. This result applies, as shown in appendix B, to semisoft response in the neoclassical model14 and appears to be general. We focused mostly on what we call the minimal model (section III), which depends only on strain and not the nematic order parameter and in which the strain tensor, _u, is constrained to be traceless. This model exhibits both semisoft and Martensitic response for a range of values of r and h that far exceed those rc and hc at the zero-stress mechanical critical point. Armed with the global phase diagram for the minimal model with external second PK stress, we calculated phase diagrams and elastic response for a number of generalizations of it: (i) one in which the first rather than the second PK stress is applied ((section IIIF), (ii) one in which the traceless constraint is replaced by a constraint on the volume by adding a term in the energy strongly favoring det Λ _ ) 1 (section IVA), and (iii) one in which the simple linear anisotropy is replaced by one nonlinear in strains (section IVB). Finally we studied the generalized neo-classical model (section V)14 depending on both Λ _ and the nematic order parameter Q. All of these models show similar semisoft behavior and regions of Martensitic behavior, underscoring the value of considering the algebraically simple minimal model. Acknowledgment. We acknowledge the support of the NSF grant DMR 0804900. F.Y. thanks Paul Goldbart for his providinganofficeattheUniversityofIllinoisatUrbana-Champaign, where part of this work was done. Appendix A. Observations about the Traceless Constraint. The traceless constraint used in the minimal model simplifies algebra and allows a straightforward analogy with the deGennes-Maier-Saupe model in perpendicular electric and magnetic fields. It does, however, present some problems that are not immediately obvious. To better understand, the implication of the traceless constraint, it is instructive to consider a model in which tracelessness is imposed by a large compressibility modulus B. In the simplest version of such a model, the free energy is expressed as a function of the traceless part of uij,
1 u˙ij ) uij - δijukk 3 and the trace of uij, φ ) ukk:
(74)
Semisoft Elastomers
1 f(uij) ) ˙f(u˙ij) + Bφ2 2
J. Phys. Chem. B, Vol. 113, No. 12, 2009 3869
(75)
In the limit B f ∞, φ is forced to be zero and u˙ij becomes equal to uij. The equation of state for uij is
∂f ∂f˙ ) + Bδijφ ) σij ∂uij ∂u˙ij
(77)
Thus in the simple model where there are separate energies for φ and u˙ij, the traceless part of uij is determined entirely by the traceless part of σij. It is useful to choose a particular representation for u˙ij. For example, using the representation of eq 19 and S ) uzz - (uxx + uyy)/2, η1 ) (uxx - uyy)/2, we obtain
∂f˙ 1 ∂f˙ 2 1 1 ∂f˙ )) σ˙ xx ) σxx - h + ∂uxx 2 ∂S 2 ∂η1 3 3
(78)
∂f˙ 1 ∂f˙ 1 1 1 ∂f˙ )) σ˙ yy ) - σxx - h (79) ∂uyy 2 ∂S 2 ∂η1 3 3 ∂f˙ ∂f˙ 2 1 ) ) σ˙ zz ) h - σxx ∂uzz ∂S 3 3
(80)
To obtain the final forms on the right-hand side of these equations, we have taken ˙f to be fiso, we included hij ) hδizδjz in σij ) hij + σxxδixδjx, and we used eq 77 to express σ˙ ij in terms of σij. These equations along with the equation for η2 are identical to eq 21. We used an alternative representation of u˙ij in eq 56 and 57 in which u˙xx ) uxx, u˙zz ) uzz, and u˙yy ) -uxx - uzz. As long as the traceless constraint is strictly applied, u˙ij and uij are equal. However, if the constraint is not strictly applied then this parametrization can be used with u˙xx ) uxx - 1/3φ, and u˙zz ) uzz - 1/3φ the independent variables and u˙yy ) -u˙xx - u˙zz. The elastic constants in eq 56 and 57 are not the constants measured in experiments. The elastic constant tensor in the simple model introduced above can be obtained by taking the second derivative of eq 75 with respect to uij
Cijkl )
∂2f ∂2˙f ) + Bδijδkl ∂uij∂ukl ∂u˙ij∂u˙kl
(
Λ 0 δ -1/2 Λ 0 0 Λ _n ) 1 -1/2 Λ1 /Λ 0 0
(76)
Using the fact that ∂f˙/∂u˙ij is traceless, we can solve for φ ) σkk/(3B), and then
∂f˙ ∂f˙ 1 ) ) σ˙ ij ≡ σij - δijσkk ∂u˙ij ∂uij 3
within the minimal model. In the treatment of the ref, 1, 14, the deformation tensor Λ _ n is measured relative to the state at σxx ) 0 rather than with respect to an isotropic state. On the plateau, it can be written as
(81)
It is clear that Cxxxx has a component proportional to B, whereas the elastic constant Cxx in eqs 56 and 57 has not. The latter elastic constant is really the rigidity with respect to the component u˙xx of u˙ij. Further details of the properties of elastic constants in semisoft systems will be presented in ref 42. B. Semisoft Strain in the Neoclassical Model. In this appendix, we will show that the strain relative to the state at the edge of the semisoft plateau calculated within the Neoclassical model of refs 14 and 1 is identical to that, eq 50, calculated
)
(82)
where Λ1 is the value of the deformation along x at the beginning of the plateau where there are no off-diagonal terms _ n- ) and δ ) 0. Thus, at the beginning of the plateau, Λ _n ) Λ -1/2 -1/2 diag[Λ1, Λ1 , Λ1 ], where diag[a, b, c] refers to a diagonal matrix with the entries a, b, c. The deformation Λ _ n′P relative to _ n(Λ _ n-)-1, and in the the plateau reference state is then Λ _ n′P ) Λ xz plane
1 _un ) [(Λ _ P - δ _] _ p)TΛ 2 2 Λ-1/2 Λδ 1 (Λ/Λ1) - 1 1 ) 2 2 Λ-1/2 Λδ (Λ1 /Λ) + Λ1δ2 - 1 1
(
)
(83)
Equating this to eq 50, we find cos 2θ ) [F + 1 - 2(Λ/Λ1)2]/ (F - 1) and
δ2 )
(Λ2 - Λ21)(FΛ21 - Λ2)
(84)
FΛ2Λ31
which is identical to eq (7.43) in ref 1. (Note, however, that here θ is the angle of rotation in the reference space, whereas θ in ref 1 is the angle of rotation of the director in the target space. These two angles are not the same, though they both reduce to 0 and π/2 at the two ends of the semisoft plateau. (For a detailed discussion of this difference in the soft case, see ref 35). In ref 1, F ) 〈l|/l⊥〉 is the average of the ratio of the parallel and perpendicular step lengths in the step-length tensor 2 /Λ_. l In the present paper, F ) (Λzz xx) is the ratio of the square of the deformations along z and x at the beginning of the plateau measured relative to the isotropic reference state. The neoclassical model discussed in refs 1 and 14 does not retain memory of the isotropic phase, so we have not verified that these two interpretations of F are equivalent, though they presumably are. It is a straightforward exercise to verify that u′nzz is equal to -1/4 (1 - F-1)(1 - cos 2θ). Thus _un′ and u_′P in eq 50 are identical in form. Let Λ _ 0 ) diag [Λ0⊥, Λ0⊥, Λ0|] be the deformation tensor at σxx ) 0. Then the deformation tensor relative to the isotropic _ 0, where Λ _ n is given by eq 82, so that at the state is Λ _) Λ _ nΛ boundary of the semisoft plateau, Λ- ) diag[Λ0⊥Λ1, Λ0⊥ 2 2 Λ1-1/2, Λ0|Λ1-1/2] and F ) (Λ0| /Λ0⊥ )Λ1-3. Then
(
2 Λ _ TΛ _ ≡ g ) Λ0⊥ Λ21
√F
Λ2 Λ21
0
0
Λ-3 1
Λ δ Λ1/2 1
0
√F
F
[
Λ δ Λ1/2 1 0
Λ21 2
Λ
]
+ Λ1δ2
)
(85)
3870 J. Phys. Chem. B, Vol. 113, No. 12, 2009
Ye and Lubensky
From this, we can calculate the traceless part of the strain tensor, _Tr g), relative to the isotropic state: u_˜ ) 1/2(g - 1/3δ
(
1 2 2 _u˜ ) 2 Λ0⊥Λ1 1 1 - s - (F - 1) cos 2θ 3 2
0
1 (F - 1) sin 2θ 2
0
2 s 3
0
1 (F - 1) sin 2θ 2
0
)
1 1 - s + (F - 1) cos 2θ 3 2 (86)
where s ) -(1 - Λ1-3) - 1/2(F - 1) < 0 because F > 1 and Λ1 > 1. Thus _u˜ has identically the same form as eq 45. At θ ) 0 2 Λ12 diag[ -1/3s, -1/2(F - 1), 2/3S, -1/3s + 1/2(F u_˜ ) 1/2Λ0⊥ 1)], is diagonal with three distinct entries, and is thus biaxial, for 0 < (F - 1) < 2|s|. For (F - 1) < 2/3|s| (i.e., F < 2 - Λ1-3), the maximum magnitude eigenvalue is along ey as it is near the transition to the high-temperature oblate uniaxial phase in the minimal model. When, (F - 1) > 2/3s, the maximum eigenvalue is 1/3s + 1/2(F - 1), and its associated direction is along ez when θ ) 0. As long as Λ1 > 1, (F - 1) < 2|s| and the system is biaxial. Thus as in the minimal model, there is biaxial order relative to an isotropic reference system throughout the semisoft phase. Perhaps not surprisingly, _u˜ has the same form as eq 86 in its uniaxial limit, F - 1 ) 2|s|. If Λ _ ) diag[Λ0⊥, Λ0⊥, Λ0|] in the uniaxial nematic state when h ) σxx ) 0,
(
)
1 3 3 - cos 2θ 0 sin 2θ 2 2 2 1 2 0 -1 0 _u˜ ) 6 Λ0⊥(F - 1) 3 1 3 sin 2θ + cos 2θ 0 2 2 2 (87) 2 2 2 /Λ0⊥ . When θ ) 0, this reduces to 1/6Λ0⊥ (F where F ) Λ0| 1)diag[ -1, -1, 2], which is clearly uniaxial. C. Elastic Moduli in the Minimal Model. In this appendix, we will calculate general expressions for the important elastic moduli of the minimal model. [See appendix A and ref 42 for comments on the meaning of these elastic constants]. These moduli are the second derivatives of the free energy density f with respect to the independent components of uij, which is understood to satisfy the modified incompressibility constraint Tru _ ) 0. To calculate these moduli, it is convenient to express u_ as
u_ )
(
uxx uxz 0 0 -uxx - uzz 0 uxz uzz 0
)
2 Czz ) 2(r + 3uxx + 12u2xx + 24uxxuzz + 24uzz + 8u2xz) (90) 2 Cxz ) r + 6(uxx + uzz) + 24u2xx + 48uxxuzz + 24uzz + 8u2xz (91) 2 C5 ) 2[r - 3(uxx + uzz) + 8(u2xx + uxxuzz + uzz ) + 24u2xz] (92)
These can be evaluated at points Pz ) (r, h, 0), where uxz ) 0, uxx ) -1/3Su(r, h), and uzz ) 2/3Su(r, h), and PY( where uxz ) 0, uxx ) -1/3SB(r, -h) ( ηB(r, -h), and uzz ) -1/3SB(r, -h) ηB(r, -h). The results are
[
Cxx(Pz) ) 2 r + 2Su(r, h) +
[
Cxx(PY() ) 2 r - SB(r, -h) +
Using this form in eq 17, taking the appropriate second derivatives, and using the notation of eq 56, we obtain, 2 Cxx ) 2(r + 3uzz + 24u2xx + 24uxxuzz + 12uzz + 8u2xz) (89)
]
(93)
20 2 S (r, -h) - 3ηB(r, -h) 3 B
]
8SB(r, -h)ηB(r, -h) + 12ηB2(r, -h) (94)
[
Czz(PZ) ) 2 r - 2Su(r, h) +
[
Czz(PY() ) 2 r - SB(r, -h) +
20 2 S (r, h) 3 u
]
(95)
20 2 S (r, -h) ( 3ηB(r, -h) 3 B
]
8SB(r, -h)ηB(r, -h) + 12ηB2(r, -h) (96)
8 Cxz(PZ) ) r + 2Su(r, h) + S2u(r, h) 3 Cxz(PY() ) r - 4SB(r, -h) +
(97)
32 2 S (r, -h) 3 B
(98)
8 h C5(PZ) ) 2 r - Su(r, h) + S2u(r, h) ) 2 (99) 3 Su(r, h)
[
]
8 C5(PY() ) 2 r + 2SB(r, -h) + SB2(r, -u) + 8ηB2(r, h) ) 0 3
[
]
(100)
Thus, in accord with the prediction of the Ward identity, C5(PZ) ) 2h/Su, and C5(PY() ) 0. On the semisoft plateau itself,
C5 ) 32Vu2xz
(88)
24 2 S (r, h) 9 u
(101)
where we return to unrescaled units. D. A Phase Boundary of SY Surface in a Real Incompressible System. We will follow a procedure similar to that used in the minimal model to determine the phase boundary of SY in the real incompressible system. Since only the phase diagram on the Y-plane (h ) σxx plane) will be investigated, the Gibbs energy density can be written as
1 g ) B[det(1 + 2u _t) + huyy (102) _) - 1]2 + fiso(u 2
Semisoft Elastomers
J. Phys. Chem. B, Vol. 113, No. 12, 2009 3871
The degeneracy possessed by the various biaxial states on the SY surface enables us to ignore the off-diagonal components of the strain tensor _u and treat _u as one having diagonal elements only. For the high-temperature equilibrium state, we have uxx0 ) uzz0 due to the symmetry on the xz plane, and the equilibrium values uxx0 and uyy0 are determined by the following equations,
∂g(uxx, uyy) )0 ∂uxx
(103a)
∂g(uxx, uyy) )0 ∂uyy
(103b)
The explicit form of these equations in term of uxx and uyy can be obtained easily in Mathematica and is not shown here to save some space. At low temperature, the rotational symmetry on the xz plane is broken, and thus uxx * uzz. To study the deviation away from the high-temperature equilibrium state, we write _u as
u_ )
(
u0xx + δS + η
0
0
u0yy + δuyy
0
0
0
u0xx + δS - η
0
)
(104)
Note we now need three independent variables, η, δS, and δuyy, rather than two since _u is no longer traceless. The Gibbs energy deviation is then a function of η, δS, and δuyy. To the fourth order of η, we have
δg ) a1η2 + a2δS2 + a3δu2yy + a4δSδuyy + a5η2δS + a6η2δuyy + a7η4 + a8η2δS2 + a9η2δu2yy + a10η2δSδuyy (105) where the coefficients ai are complicated functions of u0xx and u0yy and of other parameters as well and can be determined by Mathematica. There is no odd part of η in δg due to the symmetry between + η and -η, and the third and higher order terms of δS and δuyy have been ignored since they produce sixth and higher order terms of η. Integrating out δS and δuyy from eq 105 yields
δg ) a1η2 + C4η4 + O[η6]
(106)
with
C4 )
a3a25 - a4a5a6 + a2a26 - 4a2a3a7 + a24a7 a24 - 4a2a3
(107)
a1 ) 0 determines the second-order part of the phase boundary of SY. The tricritical point is determined by the requirement that both a1 and C4 are zero. References and Notes (1) Warner, M.; Terentjev, E. M. Liquid Crystal Elastomers; International Series of Monographs on Physics; Oxford University Press: Oxford, 2003. (2) de Gennes, P. G. Phys. Lett. 1969, 28A, 725–726. (3) de Gennes, P. G. C.R. Acad. Sci., Ser. B 1975, 281, 101–103.
(4) de Gennes, P. Weak Nematic Gels. In Liquid Crystals of One- and Two-Dimensional Order and Their Applications; Helfrich, W., Heppke, G., Eds.; Springer-Verlag: Garmishc-Partenkirchenm, Germany, 1980; pp 231237. (5) Polymer Liquid Crystals; Ciferi, A., Krigbaum, W., Meyer, R. B., Eds.; Academic Press: New York, 1982. (6) de Gennes, P.; Hebert, M.; Kant, R. Macromol. Symp. 1997, 113, 39–49. (7) de Gennes, P. G. Polym. AdV. Technol. 2002, 13, 681–682. (8) de Gennes, P. G.; Okumura, K. Europhys. Lett. 2003, 63, 76–82. (9) Warner, M.; Bladon, P.; Terentjev, E. M. J. Phys. II (France) 1994, 4, 93–102. (10) Olmsted, P. D. J. Phys. II (France) 1994, 4, 2215–2230. (11) Golubovic, L.; Lubensky, T. Phys. ReV. Lett. 1989, 40, 2631–2634. (12) Kupfer, J.; Finkelmann, H. Macromol. Chem. Phys. 1994, 195, 1353–1367. (13) Verwey, G. C.; Warner, M.; Terentjev, E. M. J. Phys. II (France) 1996, 6, 1273–1290. (14) Verwey, G. C.; Warner, M. Macromolecules 1997, 30, 4189–4195. (15) Warner, M. J. Mech. Phys. Solids 1999, 47, 1355–1377. (16) Kupfer, J.; Finkelmann, H. Makromol. Chem. Rapid Commun. 1991, 12, 717–726. (17) Finkelmann, H.; Kundler, I.; Terentjev, E. M.; Warner, M. J. Phys. II (Paris) 1997, 7, 1059–1069. (18) Kundler, I.; Finkelmann, H. Macromol. Chem. Phys. 1998, 199, 677–686. (19) Conti, S.; DeSimone, A.; Dolzmann, G. J. Mech. Phys. Solids 2002, 50, 1431–1451. (20) Bhattacharya, K. Microstructure of Martensite: Why It Forms and How It GiVes Rise to the Shape-Memory Effect; Oxford University Press: New York, 2003. (21) Ward, I. M.; Sweeney, J. An Introduction to the Mechanical Properties of Solid Polymers; John Wiley and Sons: New York, 2004. (22) Love, A. A Treatise on the Mathematical Theory of Elasticity; Dover Publications: New York, 1944. (23) Landau, L.; Lifshitz, E. Theory of Elasticity, 3rd ed; Pergamon Press: New York, 1986. (24) Chaikin, P.; Lubensky, T. C. Principles of Condensed Matter Physics; Cambridge Press: Cambridge, U.K., 1995. (25) Marsden, J. E.; Hughes, T. J. Mathematical Foundations of Elasticity; Printice-Hall, Inc.: Englewood Cliffs, NJ, 1968. (26) Stenull, O.; Lubensky, T. C. Eur. Phys. J. E 2004, 14, 333–337. (27) Lebar, A.; Kutnjak, Z.; Zumer, S.; Finkelmann, H.; Sanchez-Ferrer, A.; Zalar, B. Phys. ReV. Lett. 2005, 94, 197801. (28) Rogez, D.; Francius, G.; Finkelmann, H.; Martinoty, P. Eur. Phys. J. E 2006, 20, 369–378. (29) Brand, H. R.; Pleiner, H.; Martinoty, P. Soft Matter 2006, 2, 182– 189. (30) de Gennes, P.; Prost, J. The Physics of Liquid Crystals; Oxford University Press: Oxford, 1994. (31) Frisken, B. J.; Bergersen, B.; Palffy-Muhoray, P. Mol. Cryst. Liq. Cryst. 1987, 148, 45. (32) Bladon, P.; Terentjev, E. M.; Warner, M. J. Phys. II (France) 1994, 4, 75–91. (33) The free energy density of an elastomer can always be written as f ) fel(u_) + fQ(Q _˜ ) + fc(u_, Q _˜ ). Here Q _˜ is the tensor constructed to transform, like u_, as a tensor under rotations in the reference space [see ref 34 for details]. Thus uijQ _˜ ij is a scalar, whereas uijQij is not because Qij transforms as a tensor under rotations in the target and not the reference space. The conversion between Q and Q _˜ is implemented with the aid of the polar decomposition theorem: Λ _ )O _Λ _ S, where Λ _ S ) (Λ _ TΛ _ )1/2 ) (δ _ + 2u_)1/2 is the symmetric deformation tensor, and O _ )Λ _Λ _ -1/2 is an orthogonal rotation S matrix whose left index transforms in the target space and whose right index transforms in the references space. The partition function for this system is 3 Z ) ∫Du_DQ _˜ exp(-F[u_˜ , Q _˜ ]/T) ) ∫Du_ exp(-Feff[u_]/T),F ) ∫d xf, is the ˜Q free energy and where Feff[u_] )-T ln ∫D _˜ exp(-F[u_, Q _˜ ]/T) depends only on u_. This energy can be expressed in terms of a Landau expansion in _u. A theory in terms of the symmetric-traceless part of _u only can then be obtained by integrating out Tru _. The integration over Q _ gives rise to a shear modulus µ that passes through zero if there is an isotropic-to-nematic transition in fQ(Q _˜ ) [see ref 35]. A theory, like the neoclassical theory, expressed in terms of Λ _ and Q _ can be converted into one in terms of _u and Q _˜ using the polar decomposition results above. (34) Stenull, O.; Lubensky, T. C. Phys. ReV. E 2006, 74, 051709/124. (35) Lubensky, T. C.; Mukhopadhyay, R.; Radzihovsky, L.; Xing, X. J. Phys. ReV. E 2002, 66, 011702/1-22. (36) Fan, C.; Stephens, M. Phys. ReV. Lett. 1974, 25, 500. (37) Ye, F.; Mukhopadhyay, R.; Stenull, O.; Lubensky, T. C. Phys. ReV. Lett. 2007, 98, 147801. (38) Riedel, E. K.; Wegner, F. J. Phys. ReV. Lett. 1972, 29, 349–352.
3872 J. Phys. Chem. B, Vol. 113, No. 12, 2009 (39) Tricritical points are characterized by three fields: the temperature, the ordering field that aligns the order parameter, and the non-ordering field that couples not to the order parameter but to another field that can make the ordered phase disappear. The classical tricritical point occurs in HeHe3 mixtures. The order parameter is the superfluid condensate wavefunction ψ, and the ordering field is the field h conjugate to it. Increasing He3 concentration tends to destroy superfluid order, and the non-ordering field is the He3 chemical potential, µ3. The T-h-µ3 phase diagram in the vicinity of the tricritical point has the same geometry as that shown in Figure 4 near the tricritical point tZ. The order parameter of the semi-soft phase SZ is η. Decreases in the uniaxial order parameter S destroy the semi-soft phase
Ye and Lubensky near and below tZ just as increases in the He3 concentration destroy the superfluid phase. Since σxxuxx + huzz ) σxxη1 + (2h- σxx)S/3, the ordering field, which induces η1 ) ηx, is σxx, and the nonordering field, which induces changes in S, is (2h- σxx)/3. (40) Blume, M.; Emery, V. J.; Griffith, R. Phys. ReV. A 1971, 4, 1071. (41) Conti, S.; DeSimone, A.; Dolzmann, G. Phys. ReV. E 2002, 66, 061710. (42) Lubensky, T. C.; Ye, F. Unpublished work, 2008.
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