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Universality and Scaling in the Disordering of a Smectic Liquid Crystal
Tommaso Bellini, Leo Radzihovsky, John Toner, and Noel A. Clark
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Supplementary Material
Note 1. An important aspect of the clean bulk SmA system is its behavior in the vicinity of the nematic-smectic A (N-SmA) phase transition, at which the 1D layering of the smectic melts upon heating into a translationally disordered but long-range orientationally ordered nematic phase (with the average molecular long axis
n). This transition exhibits precursor critical phenomena (e.g., a heat capacity anomaly, softening of the crystal, and local layered clusters in the nematic), illustrated in Web fig. 1, which have been extensively studied and are understood in many (but not all) respects using modern theories of critical phenomena. The single component liquid crystal octyloxycyanobiphenyl (8CB) exhibits isotropic (I), nematic (N), smectic A (SmA), and crystal (X) phases in the bulk as follows: {I (T
IN = 40.5°C) N (T
NA = 33.5°C) SmA (T
AX = 20°C) X}. The N - SmA transition of 8CB is second order (or possibly weakly first order) as determined by x-ray and light scattering and calorimetry, as summarized in (
1). The SmA phase is heralded in the N phase by smectic-like fluctuations with power law divergent behavior versus t
|T - T
NA|/T
NA 
T/T
NA: correlation lengths,
IN along
n and
N normal to
n (red lines in Web fig. 1); heat capacity
C
pNA (black line in Web fig. 1); and susceptibility for smectic order
(T). For T < T
NA the smectic order parameter grows continuously as
t
(green line in Web fig. 1), and fluctuations are increasingly quenched for increasing
T (
S t
-V). For 8CB,
= 0.30,
= 0.19,
= 0.67,
= 0.51,
= 1.26. The N-SmA transition as currently understood differs from the well known 3D-XY model (a 3D lattice of spins free to rotate in a plane) only because of the coupling of the director degrees of freedom to the smectic order parameter. The N-SmA approaches 3D-XYcritical behavior, for which the exponents are
= -0.01,
= 0.35,
= 0.67,
= 1.32, in materials where this coupling is weak (
1).
Note 2. Smectics having fluid layers are the most fragile crystal structure that can be made in 3D space, the smectic positional ordering (1D layering) being at its lower marginal dimensionality. Thus, the relative displacement of layers due to thermally induced layer undulations, measured at points separated by r, does not decay to zero at large separations, but rather diverges weakly as log r, leaving only "quasi long-range" order. Thus, even thermal fluctuations produce a subtle loss of long-range smectic order (2, 3, 4). The effects of thermal layer undulations which give algebraic decay of the positional correlations in the bulk SmA can be assessed relative to the effects of disorder induced undulations by comparing the bulk and aerogel lineshapes. Over the T range of interest in the smectic (0°C < T < 33°C), the bulk powder scattering lineshape is effectively the resolution function (with thermal scattering in the tails). In aerogel the scattering lineshape is broadened to at least 5X the width of the resolution, indicating that the quenched disorder is the dominant disordering effect.
Note 3. The explicit form of the bulk Landau-deGennes energy functional is (5)
FLDG = a1/2(r)1/22 + w1/2(r)1/24 + c1/2(
- iq0
n(r))(r)1/22 + c1/2
z(r)1/22 + 1/2Kijkliniknl
where Kijkl are the nematic elastic constants and a(T) > (<) 0 in the nematic (smectic A) phases. The nematic correlation lengths are N(T) = [c(T)/a(T)]1/2 and N(T) = [c(T)/a(T)]1/2, determined by the gradient term coefficients c(T) and c(T) and the inverse susceptibility for smectic layer ordering a(T).
Note 4. The SmA 1D layer ordering is defined by layer amplitude and position, relating it by symmetry to other 3D condensed phase systems also described by two-component order parameters. Systems most studied include XY ferromagnets, charge density waves, vortex lattices in type II superconductors, disordered Josephson junction arrays, magnetic domain arrays and surfaces of a crystal growing on a disordered substrate. These diverse classes of systems exhibit a broad range of common phenomenology, which can be understood on the basis of analogous continuum elastic models. Because of the SmA rotational invariance the layers exhibit spring-like resistances (proportional respectively to the compression and bending moduli, B and KS), to change in spacing zu and bend, 2u, but they exhibit no resistance to uniform reorientation or to shear, in strong contrast to other 1D crystals, such as, for example, charge density waves, which are coupled to an underlying lattice. The leading bulk terms in the "elastic" description are quadratic in the spatial derivatives of layer displacement, a consequence of the spontaneous translational symmetry breaking of the clean smectic, and a feature distinguishing the smectic.
Note 5. Although macroscopically disordered in the aerogel, the smectic is defined at a randomly chosen point r by a layer normal n and location, and by a correlation function C(r) which describes how the smectic layering in the aerogel deviates from that of a perfect smectic of wavevector q0 as one moves away from r. The Fourier transform of C(r) is the "local" structure factor, Iloc(q), which can be thought of as the scattering from a single correlated domain. The measured Ipa(q) is the result of the superposition of the Iloc(q) referred to all possible local layer orientations, zloc, equivalent to the orientational (powder) averaging occurring in scattering from a crystalline powder, and can be obtained by integration over all orientations of zloc. In the limit that qo(T) and qo(T) are sufficiently large, and the ratio / sufficiently small, the powder average of Eq. 3 reduces to simply an integral over d2q.
Note 6. The data fitting procedures are as follows. The 8CB-aerogel scattering consists of three components, the disorder and thermal terms [summing to Ipa(q)] and the background term, of which the disorder term is of primary interest for present purposes. The net scattering I(q) outside the q range where the peak is growing is nearly independent of T, for T in the bulk smectic A and nematic ranges, and below, as long as freezing (suppressed in the aerogel) does not occur. Beyond this range (i.e., below freezing or in the isotropic, the background subtly changes shape making subtraction impossible. Given this condition the background was taken to be the scattering at T = 40°C, I at the highest temperature in the nematic just below the nematic-isotropic transition. The measured I(q) were then fit to the sum of Id(q) from Eq. 4, the powder average of the thermal term of Eq. 4, the background, multiplied by a fitting constant to account for the T dependence of the background scattering intensity due to change of x-ray contrast from the volume therma; expansivity of the 8CB relative to that of silica, and a constant term to account for the T dependence of nearly q-independent scattering by density fluctuations. The latter two terms, which are adjusted effectively to match up the high and low q limits of data and background (Fig. 1B), change very little between freezing and 40°C. Once I(q) is fit the background is subtracted to obtain the powder averaged scattering from the smectic layers Ipa(q) convolved with the diffractometer resolution R(q). The Ipa(q) are broad enough relative to R(q) that the effects of the latter can be eliminated by deconvolution, which was carried out by successive convolution of the data (Ipa 
R) to obtain Ipa R R, Ipa R R R , etc., followed by polynomial extrapolation at each q back to Ipa(no convolutions). This process is illustrated in Web fig. 2. The deconvolved Ipa(q), shown for the 0.05 aerogel in Web fig. 3, exhibit peaks falling initially as q-1+ 
with 
1, leveling off to a much slower decay ~ q-1 , and returning to a faster q-2 decay in the far tail. We ascribe these latter two features to the thermal component, which decays more slowly than the disorder component, and thus should appear for sufficiently large |q|. At the largest |q| the q-1 tail of the thermal component is cutoff to q-2 by the background subtraction, i.e., there is remnant thermal scattering from short range layer ordering even at T = 40 °C, where the background was taken. Subtracting the background thus removes the far tails of the thermal term, giving a q-2 falloff at large q. Various fitting functions were employed to describe the thermal component, including: (i) the powder average of that in Eq. 3; (ii) the powder average of the thermal component in Eq. 3 with the term c(q )4 added in the denominator to duplicate the form of the Iloc(q) which best fits bulk 8CB x-ray data [0 < c < 0.3 in 8CB] (6); and (iii) a Lorentzian to simply model the ultimate cutoff. All three of these choices for the thermal function yielded fits to the overall Ipa(q) of comparable quality, with essentially identical parameter sets (T), (T), and (T) for the disorder component (see Web fig. 4). This is because, as is clear from Eq. 3, that the disorder component depends much more strongly on and than the thermal, and because the thermal component is small, comparable to the background, and masked by the disorder component. Thus, although the thermal term cannot be fit uniquely, the parameters of Id(q) in Eq. 4 can be accurately determined.
At this time there is a DEL theoretical prediction for the dependence of Id(q,T) on q for large q (7), but not for the full functional form of Id(q,T). Hence, obtaining the the DEL parameters requires fitting Id(q,T) to an ad hoc function. Equation 4 is chosen because: (i) it has the Lorentzian ( = 1) form predicted by the DLDG theory for T near TNA; (ii) it yields power law (q1+) tails in q with 1+ > 2, as predicted by the Disordered Elastic (DEL) model (7) in the anomalous elastic (AE) regime; and (iii) it gives the best fits among the tested functions of , , and , subject to the constraint that -1 is always the half width q at half maximum of Isl(q,T).
Note 7. The slow growth of (T) in aerogel is a consequence of the apparently rigid nanoscopic distribution of disordering achievable in the aerogel. This situation is quite different from that in random media having pores with well-defined walls, such as Millipore, in which the layer order is only thermally perturbed within the pores but completely suppressed beyond the random boundaries defined by the pore walls. The dominant component of Isl(q) observed with Millipore confinement is Lorentzian, i.e. of the form of Eq. 4 with = 1 (8). This confirms the expectation for a random porous medium of hard walls for which the disorder part of the smectic layer correlation function is calculated to be simple exponential, giving Iloc(q) for Millipore confinement also the form of Eq. 3 (9), which, in turn, powder averages to Eq. 4 with = 1. Fig. 2 shows the resulting (T) behavior found for 8CB in Millipore filters, comparing it with that of the aerogel. In the Millipore, which apparently has little disordering within the pores, as T 
TNA+, (T) rises to and saturates at its low T value set by the pore size, for T very close to TNA. This behavior is similar to that expected for finite size confinement but very different from the slow growth of correlation observed in the aerogel. The Millipore filters used are porous cellulose acetate/nitrate manufactured with various pore size for filtering of particles and macromolecular solutes from liquids. Millipore membranes have a filtering power f in the range 0.025 < f < 5 
m, f indicating the size of the largest particles that can cross the filter. An analysis of the Millipore microscopic structure via electron microscopy reveals that the average pore chord is in the range 2f < (p( < 3f. The solid fraction is in the range 0.3 < 
< 0.5, and the solid chord is close to the pore chord. In the Millipore the conjugate interconnected networks of pore and solid are separated by a surface which is smooth on the length scale of the pores, in contrast with the aerogel where this surface is fractal on the length scale of the pores.
Note 8. Although, in the case of the smectic, and n are physical degrees of freedom to which disorder can therefore directly couple via the g(r) and V(r), their respective superconductor analogs, the Cooper pair amplitude and the electromagnetic vector potential A, are not gauge invariant objects. Thus in the quantum systems the phase of is not an observable and the coupling to g(r) is forbidden at long scale. Consequently, in superconductors and superfluids, the most important coupling allowed by gauge symmetry is the random shifts in the transition temperature Tc, introduced into the Landau energy functional of Eq. 1 by a term of the form Tc(r) ||2 (10). The transition survives such weak disorder, as evidenced by its robustness for superfluid He in low-density aerogels (11). In a smectic the effects of such random Tc disorder are negligible relative to those of g(r) and V(r) and need not even be included in the description of low-density-aerogel-induced disorder.
Note 9. The effective random-field XY free energy functional is FDLDG = a||2 + |C
|2 -Re(V(r) ). This functional gives Iloc(q)DLDG of Eq. 3 with renormalized coefficients Athermal = kBT/a(T), Adisorder V/a(T)2, and , C, (T)/xxx (T) that depend only on V (12). The renormalization a(T) is always finite and C(T) is in general T-dependent. This DLDG-based approach predicts a variety of key structural and thermodynamic features in good agreement with the experimental results presented for T near TNA, as follows: (i) Isotropic T dependence of R(T) = (T)/ (T) for T 
TNA. The coefficients a(t(T)) and C,(t(T)) renormalize to become functions of an effective reduced temperature t(T) that remains finite at all T. Thus thermal, , , and all remain finite, with C, both becoming T-independent for T TNA [case III in ref (12)]. As a result (T) (T) for T TNA. In case III the upper bound on the ratio (T)/(T) is consistent with the experimental limit (T)/(T) < 3, meaning that the powder (measured) (T) should be compared with the predicted , as is done in connection with Eqs. 3 and 4. (ii) The behavior of vs. T and V for T near TNA. The DLDG predicts (T), and, therefore (T), to depend on the single parameter V-1, which is proportional to the asymptotic slope d(T)/dT as grows with desreasing T (V-1 S d(T)/dT for T TNA). The predicted DLDG (T) were thus fitted to the (T) data by varying the single parameter V. with the results shown in Fig. 3 (light blue curves). The data show that even for T > TNA, where thermal disordering is dominant, there is significant disordering due to aerogel, as evidenced by the depression of from the bulk nematic for T > TNA. This remarkable feature, and its weal dependence on disor der in the case III regime, is correctly captured by the model. The values of V obtained (12) are consistent with the disorder dominance of the disorder term in Isl(q). (iii) The scaling of the intensity among the different aerogels. The slope S scales with the low T limit of (T) approximately as S low, so we expect that V low-1. and low Vlow2 low. This scaling among the different aerogels is qualitatively evident in Fig. 2. The DLDG fits to (T) in the different aerogels (Fig. 3) yield V ( 
, where is the aerogel solid fraction and = 1.6, whereas from the (T) data we find low(T) ( -1.5.. Thus low ( V-1, consistent the DLDG wherein the linear growth rate of (T) scales roughly as V-1 ( low. Given that alow ( low-2, we have low low. The simplest microscopic models of disorder predict = 1, a discrepancy that is currently not understood. (iv) Dominance by the (Lorentzian)2 disorder (V) term. As the correlations grow the disorder term comes to dominate Isl(q) for q < -1, consistent with the V values that fit the (T) data. Because the disorder term has a T dependence that is much stronger than that of the thermal term , growing in Eq. 3 as 1/a(T)2 versus as 1/a(T), it also controls the T dependence of the fitted (T) (see also Note 7). This is (Fig. 2). (v) The scaling behavior (T) 2(T) for T TNA in each aerogel. This is an inherent property of the V term of Eq. 3, given that Adisorder(T) = V/a(T)2 and (T)2 1/a(T). (vi) The rounding and shift to lower T of Cp with increasing is also exhibited by the DLDG model (12).
Note 10. The fits to Isl(q) for the = 0.05 and 0.10 aerogels are unquestionably improved at low T by allowing to vary, in which we find > 1 at low T. Additionally, the (T) data for all aerogels also indicate that is temperature-dependent, with > 1 at low T. Although Fig. 4 shows that these two data sets produce similar trends in , several factors complicate their detailed comparison: (i) The parameters , , and obtained will depend on the Isl(q) fitting function chosen. For the reasons stated in Note 2 we believe that Eq. 4 is the best choice Isl(q), but the parameters obtained are based on that choice; (ii) The tails of Isl(q) are predicted to have distinct regimes of different power law decay versus q. As (T) decreases with increasing T or aerogel density, the interval of q, which determines the exponent of the tail of Isl(q), shifts to higher q, potentially moving the tail of Isl(q) out of the anomalous elastic regime (7), which may account for the differences in among the aerogels at low T. This will be discussed in a future publication.
Note 11. In the weak disorder limit the liquid crystal symmetry is that of a nematic with quenched disorder, ordered locally as smectic on scales smaller than (T), but topologically disordered, with free topological layering defects (unbound dislocations) proliferating at long length scales (Fig. 1C). Dynamical restructuring of the layer system requires the motion of free dislocations and, thus, is correlated over distances much larger than (T) (13), extending to the mean distance between dislocations, which ultimately decouple layer displacement in neighboring regions. The increasing rigidity as T is lowered produces weaker local layer pinning and thus faster dynamics because of the averaging of more pinning sites within larger correlation volumes 3, as well as slower dynamics, because of the expulsion of free dislocations and a corresponding increase in the dynamical correlation length. The slowing is by far dominant, the apparent divergences of the stretching times indicating the divergence of the dynamical correlation length, and the approach to a glassy state. The theory shows that the SBG is stable against sufficiently weak disorder if and only if 
B and K (both > 0) satisfy the inequalities B + K < 2 and K < 1 (13). Because B + K 4/(+2) is about 1.55, B < 2 is clearly satisfied. Because B,K > 0, and we expect that B > K for D = 3 (based on the exact calculation for D = 5, where B 
5K), it is almost certain that K < 1 is satisfied. Thus the theoretical treatment based on the DEL model combined with the experimental result strongly indicates that the SmBG should be achievable in the 8CB aerogel system. The fact that it is apparently only approached suggests that the disorder in these particular experiments is simply too strong, and that in experiments with yet weaker disorder it will be possible to observe the SmBG. In this case the nematic - SmBG transition should by marked by a true (unrounded) specific heat singularity and a that remains finite.
References
1. C. W. Garland, G. Nounesis, Phys. Rev. E 49, 2964 (1995).
2. A. Caille, C. R. Acad. Sci. Ser. B 274, 891 (1972).
3. J. Als-Nielsen et al., Phys. Rev. B 22 312 (1980).
4. C. R. Safinya et al., Phys. Rev. Lett. 57, 2718 (1986).
5. P. G. deGennes, J. Prost, The Physics of Liquid Crystals (Oxford, London, 1993).
6. D. Davidov et al., Phys. Rev. B 19, 1657 (1979).
7. L. Radzihovsky, J. Toner, unpublished theory.
8. A. Rappaport, G. M. Danner, B. N. Thomas, N. A. Clark, in preparation.
9. T. Bellini, N. A. Clark, in Liquid Crystals in Complex Geometries, G. P. Crawford, S. Zumer, Eds. (Taylor and Francis, London 1996), chap. 19.
10. K. Moon, S. M. Girvin, Phys. Rev. Lett. 75, 1328 (1995).
11. M. Chan, N. Mulders, J. Reppy, Phys. Today August, 30 (1996).
12. B. Ward, Ph.D. Thesis, University of Colorado (1999); L. Radzihovsky, B. Ward, unpublished data.
13. L. Radzihovsky, J. Toner, Phys. Rev. B 60, 206 (1999).
Supplemental Figure 1. Critical behavior of 8CB near the second order nematic-smectic A transition. The insets sketch the smectic molecular ordering (T < TNA- green background) of 2D liquid layers (spacing d) with its associated modulation of the density ((z) along the layer normal z, and the nematic ordering (T > TNA- yellow background) with mean molecular long axis (director) orientation n. TNA = 33.5°C is the transition temperature. The SmA phase is heralded in the N phase by smectic-like fluctuations with power law divergent behavior versus t |T - TNA|/TNA T/TNA: correlation lengths, N along n and N normal to n(red lines); heat capacity CpNA (black line); and susceptibility for smectic order (T) (6). For T < TNA the smectic order parameter grows continuously as t(green line), and fluctuations are increasingly quenched for increasing T (S t-V). For 8CB, = 0.30, = 0.19, = 0.67, = 0.51, = 1.26. The N-SmA transition as currently understood differs from the well known 3D-XY model (a 3D lattice of spins free to rotate in a plane) only because of the coupling of the director degrees of freedom to the smectic order parameter. The N-SmA approaches 3D-XYcritical behavior, for which the exponents are = -0.01, = 0.35, = 0.67, = 1.32, in materials where this coupling is weak (1).
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Supplemental Figure 2. Deconvolution of the diffractometer resolution R(q) from Ipa(q) for = 0.05, T = 5.4 °C, where the effects of R(q) are the strongest. Shown are the data (black line), corresponding to one convolution of R(q) with the actual lineshape, the data with several successive convolutions, and the extrapolation back to zero convolutions, giving the resolution-free line shape.
Medium version | Full size version
Supplemental Figure 3. Log-log plots of the growth of Ipa(q) with decreasing temperature in the = 0.05 aerogel showing the overlap of the tails.
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Supplemental Figure 4. Ipa(q) data with fitted curves, assuming the thermal term to have either a Lorentzian form (black curves) or the powder average of the thermal term in Eq. 3 (magenta curves). The resulting disorder parts of Ipa(q) are shown respectively as the black and magenta dashed curves, and do not depend significantly on the form used to fit the thermal part. The inset shows the growth of the disorder component of Ipa(q) versus T, and in particular the increase in the slope of the tail, evidence for anomalous elasticity.
Medium version | Full size version
