4 Random anisotropy liquid crystal model

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Phase Transitions. Applications to Liquid Crystals, Organic Electronic and Optoelectronic Fields, 2006: 79-96 ISBN: 81-308-0062-4 Editor: Vlad Popa-Nita

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Random anisotropy liquid crystal model S. Kralj1,2, V. Popa-Nita3 and M. Svetec1 1 Laboratory of Physics of Complex Systems, Faculty of Education, University of Maribor, Koroška 160, 2000 Maribor, Slovenia; 2Jožef Stefan Institute Jamova 39, 1000 Ljubljana, Slovenia; 3Faculty of Physics, University of Bucharest, P. O. Box MG-11, Bucharest 76900, Romania

Introduction We study temperature driven phase behavior of a randomly perturbed liquid crystal (LC). We use Landau-Ginzburg-de Gennes type approach combined with the anisotropy random field model. In bulk LC exhibits discontinuous isotropic-nematic (I-N) and continuous nematic-smectic A (N-SmA) phase transition. Our approach indicates that for strong enough disorder, that is directly coupled to order parameters of the system, both phase transitions cease to exist. The N-SmA transition is apparently affected by disorder only if the translational ordering is directly coupled to the disorder. Moreover, in the regime of Correspondence/Reprint request: Dr. S. Kralj, Laboratory of Physics of Complex Systems, Faculty of Education, University of Maribor, Koroška 160, 2000 Maribor, Slovenia. E-mail: [email protected]

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relatively weak disorder strength the bulk continuous N-SmA transition becomes discontinuous. It is practically impossible to make a perfectly pure sample. Impurities, introducing a kind of disorder into the sample system, are always present. They can often dramatically alter the behavior anticipated in a pure bulk. Phases, reached via a continuous symmetry breaking phase transition, are particularly susceptible to impurities. The reason behind this is the presence of a zero gap Goldstone mode, trying to recover the lost continuous symmetry. According to the Imry-Ma theorem [1] even an arbitrarily weak uncorrelated disorder breaks such a system into a domain-like pattern in spatial dimensions d < 4. Random field type approaches [1] predict the relation ξd ∝ w

−

2 4− d

, where ξd is the

characteristic domain size and w measures the disorder strength. Therefore a kind of glass-type short range ordering is expected to appear. Some recent studies [2, 3, 4, 6] question the validity of the Imry-Ma argument. They predict that a quasi long range order (QLRO) is established instead characterized by an algebraic decay of spatial correlations. According to Giamarchi and Doussal [3] the Imry-Ma domain size is nevertheless present also in this case. Contrary to the original Imry-Ma assumption, where ξd estimates the size of more or less independent crystallites, ξd now measures the distance after which the relevant correlation function approximately reaches a finite plateau value. Studies focusing on Imry-Ma domain-type ordering of weakly perturbed systems have been so far mostly carried out in magnets and superconductors [1, 2, 3, 7]. Recently [8] it has been shown that binary systems A+B, where A stands for a liquid crystal (LC) phase and B for a chemically inert component, acting as a kind of random perturbation to A, are also suitable for such studies. The reason for this is the soft character of most LC phases and the existence of numerous aperiodic and periodic phases&structures where a continuous orientational or translational symmetry is broken [9]. As B, different porous matrices (e.g., Russian glasses, aerogels, Vycor and Controlled-Pore Glasses) or particle-like (e.g., spherular aerosils) inclusions are commonly chosen [8]. The spatial orientation and location of the A-B interface is essentially random. Because of the ability of this interface to anchor LC molecules in certain directions and to pin smectic layers, it introduces an essentially random perturbation to the LC ordering. All these systems are characterized by some geometrically imposed length R. One intuitively expects that the disorder strength decreases with increasing R, however there can be exceptions. In case of LCs there is no clear consensus about the nature of a weakly perturbed LC phase. Several light [11, 10] and x-ray [13, 12, 14, 15, 16] scattering experiments support the existence of clusters and of short range order (SRO) of perturbed LC phases. However, a recent NMR study on 8CB confined to high

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density silica gels reports on long range order [17]. In addition, calorimetric results [18] in LCs confined to aerogels, aerosils and CPGs all reveal finite size effects only in terms of the characteristic geometric size R. Therefore, if in these samples a domain pattern exists, it relatively weakly affects the fluctuation spectrum of the system. Some simulations [10] support SRO ordering, while others [4, 5] predict QLRO despite using similar (random anisotropy type) models in all cases. Detailed theoretical studies [19] have shown that disorder arising due to the pinning of orientational order at the LCporous interface always destroys long range LC ordering. For weak enough disorder the existence of Bragg-glass phases exhibiting QLRO has been predicted. This kind of ordering in the nematic phase is also predicted by Feldman [6] using a renormalisation group study. Theoretical predictions and experimental observations are however still not well matched, particularly for the case of a relatively weak disorder. In this contribution we analyze theoretically the influence of relatively weak quenched random disorder on discontinuous isotropic-nematic (I-N) and continuous nematic-smectic A (N-SmA) phase transition. In these cases the continuous orientational and translational symmetries are broke, respectively. We originate from the Landau-Ginzburg-de Gennes type phenomenological approach in combination with Random Anisotropy Field model [4]. We study phase behavior for the case i) when only orientational and ii) when in addition the translational ordering is also directly coupled to disorder. Particular focus is paid to the disorder induced crossover from continuous to discontinuous character of the N-SmA phase transition. The outline of the paper is as follows. In Sec. 4.1 we introduce the model. In Sec. 4.2 we derive the effective dimensionless free energy, resulting from the expected domain pattern. The phase behavior as a function of temperature and disorder strength is studied in Sec. 4.3. The robustness of the observed disorder induced crossover in the N-SmA phase transition is analyzed. In the last section we summarize the results.

4.1. Model We consider a binary system A+B. Here A stands for a liquid crystal and B acts as an random perturber to A. The thermotropic liquid crystal exhibits the first order isotropic-nematic (I-N) and the second order nematic-smectic A (N-SmA) phase transition as temperature is varied. The chemically inert component B enforces to LC a kind of relatively weak random disorder. For example, in most experiments the role of B is played by porous matrices or aerosil particles. We describe the LC ordering using the Landau-Ginzburg-de Gennes phenomenological description [9]. We represent the orientational ordering with

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G G the uniaxial tensor order parameter Q = S(n ⊗ n − I / 3). Therefore, biaxial effects

are neglected. Here ⊗ marks the tensorial product and I is the identity tensor. The translational ordering is described with the complex order parameter ψ = G ηeiφ. The uniaxial orientational order parameter S(r) and the translational order G parameter field η(r) reveal the degree of orientational and translational ordering, GG respectively. The unit vector field n(r) points along the local average uniaxial G orientation of LC molecules and the phase field φ (r) locates smectic layers. For G G the smectic layers stacked along the wave vector q0 the phase is given by φ(r) = G GG G G q0 ⋅ r . The nematic director n(r) and the phase φ (r) are classified as hydrodynamic fields. The nematic director field suffers the loss of continuous symmetry at the I-N and the phase field at the N-SmA phase transition. In the nematic and smectic A phase, respectively, they exhibit (non-massive) longwave modes with vanishing energy costs in the long-wavelength limit. In terms of these fields we express the free energy F of the confined liquid crystal phase, focusing on the I-N and N-SmA phase transition as (4.1) The condensation terms f c(phase) determine the I-N (f c(n)) and N-SmA (f c(s)) phase transition behavior of an undistorted liquid crystal in case of negligible coupling term fcp between the nematic and smectic order parameters. These terms are expressed as

(4.2) (4.3) (4.4) The positive quantities an, T*, bn, cn, as, bs, TNA, D1 and D2 are the material constants. They determine the bulk value of the orientational (S ≡ Sb) and translational (η ≡ ηb) order parameter. The equilibrium ordering results from = 0, corresponding to the solutions to the equations global minimum of F. By increasing D1 the N-SmA transition crosses from the second order to the first order character at the critical value D1 ≡ Dc [9]. The

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saturation term proportional to D2 allows for reentrant temperature driven nematic behavior [20]. The bulk values are for fcp = 0 determined by Sb(T ≤ , Sb(T > TIN) = 0, ηb(T ≤ TNA) = as(TNA − T) / cs, TIN) =

ηb(T > TNA) = 0. Here TIN = T* +

T*, and TNA stand for the I-N transition, nematic supercooling, and N-SmA transition temperature, respectively. The nematic (f e(n)) and smectic (f e(s)) elastic terms are expressed as (4.5) (4.6) The elastic LC properties are described with positive representative bare nematic (L), smectic bend (C⊥) and smectic compressibility (C&) elastic constants. We henceforth set C⊥ ~ C& ≡ Ls, neglecting the smectic elastic anisotropy of the system. The elastic terms weighted by elastic constant i) L, ii) C⊥, and iii) C& enforce i) homogeneous nematic orientational ordering, ii) G parallel alignment of n and the normal to a smectic layer, and iii) the layer distance d0 = 2π/q0. The essential interface free energy density contributions are expressed as [7, 19] f i = f i (n) + f i (s) : (4.7) (4.8) and W = Wseiφs. The positive quantities Wn and Ws stand for the characteristic G orientational (Wn) and translational (Ws) anchoring constant, ν is the local normal of the A-B interface, and φs is the smectic phase enforced by the interface. Note that we have introduced only the most important free energy terms that are essential for a qualitative behavior of the model.

4.2. Effective model In the following we derive the scaled effective free energy of the system, reflecting the influence of a relatively weak random quenched disorder.

4.2.1. Domain pattern We henceforth assume that the disorder transforms the LC structure into a domain-type structure. An average domain is characterized by a single average

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domain size ξd. Over this size LC hydrodynamic fields exhibit pronounced change. However, this domain structure does not resemble independent crystallites. It exhibits rather smooth variations in which a characteristic length is present. This description coincides with the one given by Giamarchi and Doussal [3] for weakly perturbed magnetic systems with QLRO. We further propose that for weak disorder the established LC pattern closely resembles a “temporal” domain structure arising in the so called Kibble-Zurek mechanism (KZM) [21, 22]. In pure systems this pattern arises following a fast enough (with respect to a relevant order parameter relaxation time) continuous symmetry breaking phase transition. This mechanism was originally introduced in cosmology to explain coarsening dynamics of the Higg's field. The main KZM ingredients are just i) continuous symmetry breaking and ii) causality (i.e., a finite speed with which information spreads in a system). Because these phenomena appear so general they can be applied also to condensed matter systems, as already demonstrated by Zurek [22]. For illustration we consider a system undergoing a temperature driven phase transition quench. After the quench in causally disconnected parts of the system a different value of the symmetry breaking order parameter value is generally chosen. Consequently, a domain structure appears characterized by a single characteristic length ξD(t). Subsequent domain growth follows a scaling law ξD ∝ tζ, where ζ stands for the scaling coefficient and t measures the time after the quench. If impurities are present they could pin the domains causing a frozen domain pattern, where the established final value ξD(t = ∞) ≡ ξd depends on concentration and pinning capabilities of impurities. Note also that in case of impurities the domain pattern could arise also in relatively slow quenches because of spatially different initiatives for a symmetry breaking direction.

4.2.2. Typical elastic distortions We replace local order parameter field values with their spatial averages. G G Thus S ~ < S(r) >, η ~ < η(r) >, where stands for the spatial average over the LC domain. In addition we take into account that the distorted LC hydrodynamic fields in the absence of external electric or magnetic fields typically evolve over the geometrically imposed scale. In our case this scale is given by ξd. Therefore within an average domain it holds true (4.9)

JJG G where ϕ = φ − q0 ⋅ r.

(4.10)

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We further, in the spirit of Random Anisotropy Field approach [23, 24], assume that via essentially randomly geometrically varying A-B interface LC orientational (i = n) and translational (i = s) ordering suffers Ni ~ (ξd/ri)3 random reorientations within the three dimensional domain volume Vd. The length ri measures a typical distance along which the i-th degree of freedom experiences a substantial random change. It roughly coincides with a typical G length on which the orientation of ν apparently changes. Thus Nn and Ns count a number of random changes that the orientational (Nn) and translational (Ns) ordering suffers in the average domain. In estimating the averaging effects that the interface terms introduce into the system we rely on the central limit theorem according to which (4.11) (4.12)

4.2.3. Effective scaled free energy Taking this into account we express the average free energy density f =< Fd > /Vd of the system where < Fd > stands for the average free energy of LC within Vd ~ ξd3. For numerical purpose we introduce the following scaling [20]: (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) The ratio Ad/Vd measures the average area Ad of the LC-perturber interface in Vd. In order to avoid singularity for ξd = 0 we further introduce the dimensionless lengths

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(4.19) We henceforth refer to ξn and ξs as the nematic and smectic relative domain correlation length, respectively. Note that these distances are measured in units of ri. For the Landau coefficients appearing in we use in calculations the values cn = 5.0 × 107 bn = 2.0 × 107 given in [25]: an = 0.1 × 107 as = 0.05 × 107 cs = 0.4 × 107 D1 = 0.225 × 107 D2 = 0.7 × 107 TIN − T* = 1.8K, TNA − T* = −8.7K. Chosen values for D1 and D2 correspond to a second order N-SmA bulk phase transition. Eliminating the overbars in Eqs.(4.15,4,16,4.17,4.18), we obtain the scaled free energy density f = fc + fe + fRF. The condensation (fc), elastic (fe), and random field (fRF) term are expressed as

(4.20) (4.21) (4.22) The fc contribution reveals the competition between ordering and thermal disordering fluctuations. The temperature dependence enforced by fc for the chosen set of parameters is shown in Fig. 4.1.

Figure 4.1. Temperature variation of the nematic (S, full line) and smectic (η, dashed line) order parameter in the bulk sample. We square the smectic order parameter in the figure in order to emphasize the square root temperature dependence of η.

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The system exhibits a first order nematic-isotropic phase transition at t ≡ tIN = 1 and a second order nematic-smectic-A phase transition at t = tNA = −4.943. The jump of the nematic order parameter at the transition is S ≡ SIN = 0.2667. The elastic term tends to establish a homogeneous ordering with ξn = ξs → ∞. The disorder terms is a source of additional static disorder. This term introduces a kind of frustration into the system. Its local tendency is to enhance the ordering because fRF ∝ −θ, where the subscript RF stands for the random field. The quantity θ represents the relevant order parameter (i.e., θ = S and θ = η for the nematic and smectic case, respectively. However fRF also enforces ξn = ξs → 0. This introduces elastic penalties into the system, that tend to decrease the degree of ordering.

4.3. Phase behavior We next study the phase behavior of the effective model. Minimizing the free energy density f with respect to ξn and ξs, we obtain the equilibrium values of the nematic and smectic relative domain correlation lengths:

(4.23) and

(4.24) Here S0 ≡ and η0 ≡ Taking this into account we obtain the dimensionless free energy density as a function of order parameters: (4.25) where

(4.26)

(4.27) In the following we study the main topological features of phase diagrams as temperature and the disorder strengths Wn and Ws are varied. For Wn = Ws = 0 the

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system can exhibit i) isotropic, ii) nematic, and iii) smectic A phase. They are characterized by i) S = 0, ξn = 0; ii) S > 0, ξn = ∞; iii) η > 0, ξs = ∞. In general four additional possible equilibrium states are encountered in the presence of disorder. These are the i) paranematic, ii) speronematic, iii) parasmectic, and iv) sperosmectic phase. The i) paranematic phase that closely resembles the isotropic phase. It exhibits a low, but finite degree (i.e., S 2 0) of orientational order and ξn = 0. The ii) speronematic phase represents a distorted nematic phase, with S > 0 and a finite value of ξn. In iii) the parasmectic phase a finite degree of translational order appears, η 2 0 and ξs = 0. The iv) sperosmectic phase is characterized by η > 0 and ξs > 0. We henceforth for simplicity, which does not affect the qualitative picture of the model, consider the elastic isotropic model in which Ln ~ Ls ≡ L = 1.

4.3.1. Orientational disorder We first consider the case in which Ws = 0 and Wn > 0. Therefore, only the orientational degree of ordering is directly coupled with the disorder. In this case the free energy density f defines the paranematic, speronematic, and smectic-A phase as the possible equilibrium states. The (Wn, t) phase diagrams is presented in Fig. 4.2. For a low degree of disorder (Wn < Wn(c) ~ 0.56) there is a first order phase transition from a paranematic phase to a speronematic phase, which is marked with the full line. At higher values of Wn (Wn > Wn(c) ) the transition between

Figure 4.2. Phase diagram in the (Wn, t) plane of the phase space. The full line defines the discontinuous transition between the paranematic and speronematic phase. The dashed line marks the continuous speronematic-SmA transition. The dotted line marks the continuous paranematic and speronematic transition and is probably the artifact of the model. In reality a strong variation of order parameter is expected as this line is crossed. The circle and square mark the tricritical ( Wn(c) ~ 0.56) and triple ( Wn(t) ~ 0.79) point, respectively.

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paranematic and speronematic becomes continuous (the dotted curve). We believe that this transition in actuality corresponds to a gradual evolution of ordering. The tricritical point at Wn ≡ Wn(c) ~ 0.56 is marked with the circle. The speronematic-smectic phase transition remains second order. The phase transition temperature tNA (the dashed line in Fig. 4.2) is only slightly increased with increased Wn. This is due to the indirect influence of Wn on η via the (S, η) coupling terms in Eq.(4.20). The three phases coexist at the triple point, taking place at Wn ≡ Wn(t) ~ 0.79 (the square in Fig.2). For Wn > Wn(t) the transition takes place between isotropic and smectic-A phases. The profiles of the nematic and smectic order parameter as well as the behavior of the nematic correlation length ξn as a function of temperature are shown in Figs. 4.3 and 4.4.

Figure 4.3. Order parameter temperature dependences for Wn = 0.3 < Wn(c) (full line) and Wn = 0.6 > Wn(c) (dashed line).

Figure 4.4. ξn as a function of temperature for Wn = 0.3 < Wn(c) (full line) and Wn = 0.6 > Wn(c) (dashed line).

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The cases below Wn(c) (i.e., Wn = 0.3; full line) and above it (i.e., Wn = 0.6) are shown. For Wn = 0.3 the paranematic-speronematic phase transition is discontinuous, see Fig. 4.3. For Wn = 0.6 the transition becomes continuous and is characterized by the discontinuity in the derivative dS/dt. This feature is most probable the artifact of the model and is in reality replaced by gradual evolution of ordering. Similarly for Wn < Wn(c) there is a discontinuity in ξn at the first order paranematic-speronematic phase transition, as it is shown in Fig.4.4 with the full line. Above the tricritical point ξn increases continuously from zero with increasing Wn (dashed curve in Fig. 4.4).

4.3.2. Orientational and translational disorder The (W, t) phase diagram for the case where both degrees of ordering directly experience disorder is presented in Fig. 4.5. For simplicity we set W = Wn = Ws. For a relatively weak disorder strength (i.e., W < W(c) ~ 0.56) there is a first order phase transition from a paranematic phase to a speronematic phase, shown with the full line. At higher values of W (W > W(c)) the transition between paranematic and speronematic becomes continuous (the dotted curve). Here W = W(c) ~ 0.56 defines the tricritical point. Therefore, finite value of Ws does not introduce qualitative changes to the paranematic-speronematic phase transition line. However, the speronematicsperosmectic phase transition (the dashed line in Fig.5) becomes first order. The smectic order parameter experiences a jump at the transition, as it is shown in Fig. 4.6.

Figure 4.5. Phase diagram in the (Wn, t) plane of the phase space for Wn = Ws ≡ W. The full line defines the discontinuous transition between the paranematic and speronematic phase. The dashed line marks the discontinuous speronematic-sperosmectic transition. The dotted line marks the continuous paranematic and speronematic transition and is probably the artifact of the model. The circle and square mark the tricritical ( Wn(c) ~ 0.56) and triple ( Wn(t) ~ 1.03.) point, respectively.

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Figure 4.6. Order parameters temperature dependencies for Wn = Ws ≡ W = 0.3 < Wn(c) (full line) and W = 0.6 > Wn(c) (dashed line).

Figure 4.7. ξn as a function of temperature for Wn = Ws ≡ W = 0.3 < Wn(c) (full line) and W = 0.6 > Wn(c) (dashed line).

The same holds true for the ξs(T) dependence, see Fig. 4.7. The transition temperature line tNA(W) decreases with increasing W. The three phases coexist at the triple point defined by W ≡ W(t) ~ 1.03.

4.3.3. Disorder and critical behavior Our analysis shows that a strong enough disorder can change a character of a phase transition. In particular we see, that the second order N-SmA phase transition becomes discontinuous for Ws > 0. This result is rather surprising, so we treat it in a more detail. We carry out a comparative study how random disorder influences the 1st and 2nd order phase transition in which a continuous symmetry is broken. For this purpose we study the effective free energy of the form

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(4.28) where θ stands for the order parameter and ξ is the domain correlation length. The condensation term f c(i) triggers alone either the 1st (i = 1) or 2nd (i = 3) order phase transition at the transition temperature t ≡ tc. We express it as (4.29) (4.30) We refer to the t > tc and t < tc phase as the isotropic and ordered phase, respectively. The equilibrium phase behavior of f c(i) is the following. For the first order transition we have θb(t > tc) = 0 and θb(t < tc) =

tc = 0.

The supercooling (t ≡ t*) and superheating (t ≡ t**) values of t equal to t* = −1 and t** = 1/8, respectively. For the second order transition the equilibrium order parameter equals θb(t > tc) = 0 and θb(t < tc) = −t ; tc = 0. We proceed by studying the phase behavior of Eq.(4.28). Minimizing f(i) with respect to ξ leads to

(4.31) where θ0 = 3W/(4L) stand for the crossover value of the order parameter. The resulting free energy can be now expressed as a function of θ only:

(4.32) The subscripts p and s stand for the paraorder and speroorder phase, respectively, referring to the phase ordering for W > 0. The paraorder phase refers to a weakly ordered isotropic phase with ξ = 0. The speroorder phase represents an ordered phase with a finite value of ξ. We study numerically phase transition between paraorder and speroorder phase, taking place at tc = tc(W, L). The resulting phase diagram in the (W, t) plane, calculated for L = 1, is shown in Fig. 4.8. In Fig. 4.9 we plot the jump ∆θ of the order parameter at transition lines, measuring the strength of the first order phase transitions. The average size of domains is shown in Fig. 4.10.

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Figure 4.8. The transition temperature tc as a function of W for the continuous (dashed line) and discontinuous (full line) bulk phase transition.

Figure 4.9. Jump of the order parameter ∆θ on crossing tc as a function of W.

Figure 4.10. Dependence of ξ on W.

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In both cases with increased strength W the critical vale tc(W) monotonously decreases if W < Wc(i) , where Wc(1) ~ 1.39 and Wc(2) ~ 1.40. Above the critical strength Wc(i) the temperature derivative of the order parameter exhibits discontinuity, which is the artifact of our approach. We emphasize that the disorder converts the 2nd order transition into a discontinuous one, provided that 0 < W < Wc(2) ~ 1.40. The first order transition character for W > 0 steams from the competition between the condensation (f c(2)) term, which is ready to establish an ordered state for t < 0 and the disordering term (fRF). In the regime 0 < W < Wc(2) these terms are comparable. Within the interval [0, Wc(2) ] there are two distinct regimes. For a weak enough disorder (W < 0.4) the strength of transition is the increasing, and above this regime the decreasing function of W. For W > Wc(2) the noncritical disorder contribution is strong enough to wash out discontinuity in ∆θ as t is varied.

4.4. Conclusions We study theoretically the influence of a relatively weak quenched disorder on the phase behavior of a LC. We focus on the I-N and N-SmA phase transition. We use combined Landau-Ginzburg-de Gennes and random anisotropy field-type approach. General features emerging from this model adequately mimic to some extent relevant characteristics of a perturbed nematic or SmA phase, where the role of the perturber is played either by the CPG matrix or aerosil spherular particles. In these systems the disorder is enforced geometrically via essentially random variation of easy-axes orientations at the LC-perturber interface. We further predict that the sample exhibits a domain-like pattern, characterized by an average domain length ξd. This prediction is sensible in cases where a continuous symmetry is broken at a phase transition. For I-N and N-SmA phase transition the continuous orientational and translational symmetry is broken, respectively. In addition the disorder strength should be weak enough in order to enable a single, relatively sharply peaked distribution of ξd values. The domain pattern is in line with recent observations in aerosil (turbidity experiments [10], X-ray scattering [14]) and CPG (X-ray scattering [26]) samples. The proposed model suggests that for strong enough disorder the I-N and N-SmA phase transition becomes gradual. The N-SmA transition is significantly affected by the disorder only if the disorder is directly coupled to the smectic ordering. The model implies that an arbitrary weak disorder converts a secondorder N-SmA transition into a discontinuous one. Note that in the limit of strong disorder the assumption of a relatively sharply peak distribution of ξd

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values becomes questionable. Therefore, in our approach we underestimate the influence of disorder. Consequently the observed effects should be even more pronounced in a more accurate treatment. Note that we do not ask what is the true nature of the speronematic and sperosmectic phase in our approach. This is still an open problem and there exist contradicting predictions. It is believed that either the short range (SR) [27], quasi long range (QLR) [28, 6] or even long range (for weak enough disorder [28]) order is established. To study the true nature of the disordered nematic phase one should carry out a study in which fluctuations are taken into account. For example a Monte Carlo study is adequate for such a purpose, in which the orientational correlation function g(r) should be studied as a function of the separation r in a finite system characterized by a linear size R. A finite size analysis is needed to check the limiting R → ∞ behavior g(r ~ R). If this limit is finite it signals the true long range order. In opposite case the exponential decay of correlations signals the SR order and algebraic decay the QLR order. The proposed study is of relevance also for other randomly perturbed systems. The most important condition for the applicability of our approach is that a continuous symmetry is broken at the phase transition. Looking at this broader picture, we stress the following support to qualitative validity of our results. It has been shown by Imry and Wortis [29] on magnetic systems that above critical disorder strength a first order phase transitions ceases to exist. Concerning the disorder driven transformation of a continuous phase transition into discontinuous one we refer to the study carried out by Aharony [30] on randomly perturbed magnets using the renormalization group. He observed that random perturbations can destabilize a fixed point describing the continuous phase transition taking place in the pure case. He claims that this instability is the signal of a smeared transition, although the first order transition might also be the reason for it. It is well possible that in a real system both effects are present.

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