NEMATIC PHASES IN FLUIDS OF BIAXIAL PARTICLES By ANDREW CHARLES EDMUND REID B Sc. University of British Columbia 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 19S9 ANDREW CHARLES EDMUND REID , 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date -Zr?{%°[ DE-6 (2/88) 11 Abstract The investigation of the possible uniaxial phases of a fluid of biaxial particles undertaken by Bergersen, Palffy-Muhoray and Dunmur forms the starting point for further research into the full range of possible phases of such a fluid. A generalization of their interaction model, free from constraints having to do with interaction details, retaining only the biaxial symmetry, is used in the mean-field approximation to explore the range of possible orientationally-ordered phases for such a system. This model is an equilibrium model which does not include dynamic effects. The basic inter-particle interaction is abstract, having the correct symmetry for biaxial particles, and is the most general biaxial interaction constructable from lowest-order scalar invariants. The self-consistent equations resulting from this formulation are obtained in the mean-field approximation and therefore retain both the symmetry and the generality at the cost of exact numerical correctness. Four order parameters are identified, corresponding (within a numerical factor) to those found by Straley and Freiser, as well as those of Bergersen et al. The phase diagram of a fluid of biaxial particles, then, is mapped out in terms of the behavior of these order parameters, as indicated by the self-consistent equations, as a function of three anisotropy parameters and the temperature. The primary method of analysis is iteration of the self-consistent equations obtained from differentiating the free energy. Numerical results are obtained for the location of the phase boundaries, and the temperature dependence of the order parameters in various phases. Ill Table of Contents Abstract ii ^ Table of Content iii List of Figures iv „ List of Tables v Acknowledments vi Introduction 1 Theoretical Model 3 Analj'sis 13 Conclusions • 20 Figures 21 Table 28 Bibliography 29 Listing of Programs 30 I V List of Figures page Figure 1. Order parameter S as a function of temperature in the Maier Saupe model. 21 Figure 2. Order parameter C (left curve) and S (right curve) vs. temperature for u=0.3266, v=0.0266. 22 Figure 3. Order parameter C (upper curve) and S (lower curve) in the case u = ^» v = § ' ^ Figure 4. S (lower curve) and C (upper curve) vs. T for the case u=0.8447, v=0.0948, the self dual case in Straley's model. 24 Figure 5. S (right curve) and C (left curve) for u=0.6573, v=0.0506, rod-like case in Straley's model. 25 Figure 6. S (lower curve) and C (upper curve) vs T for u=0.9297, v=0.1204, plate-like case in Straley's model. 26 Figure 7. Phase diagram map. 27 V Table Caption page J Table 1: MAPPER output for u=0.5 , v=0.1 and 0.2. Negative T Q b indicates plate-like phase. 28 V I Acknowledgement The author wishes to thank Dr. Birger Bergersen for his guidance in this project, and for his tolerance and good nature. Thanks also to Peter Holds worth for many illuminating conversations in the early daj's, attempting to grasp condensed matter generally. A special thank you to Tom Nicol of the UBC Computing Centre, for knowing all that stuff. 1 Introduction The stud}' of biaxial phases in nematic liquid crystals has taken many forms since the discovery of a lyotropic biaxial nematic phase^ several years ago, and possible 2 3 thermotropic biaxial nematic phase more recently ' . A particular model for a thermotropic biaxial nematic liquid crystal, studied by Bergersen, Palffy-Muhoray and Dunmur and based on a van der Waals type interaction, forms the starting point for a more general study of biaxial phases in nematic liquid crystals presented here. The model considered is for thermotropic biaxial nematic phases of pure fluids of rigid particles in zero-field, in contrast to other models which incorporate flexibility or are about mixtures, or pure substances in crossed fields. Such models often exhibit biaxial nematic phases, but it is not necessarily clear how the interparticle interaction is involved. In this model, the symmetry of the interparticle interaction is the source of any biaxiality. Often, bulk uniaxial phases of biaxial particles exhibit biaxial effects 5 6 detectable by NMR and optical dichroism , for instance. There is also unambiguous 1 2 3 biaxiality in lyotropic , and somewhat less obvious in thermotropic, liquid crystals ' . 7 Biaxiality has also been predicted but not found in mixtures of rods and disks . The interaction provided by Bergersen et al^ is generalized beyond it's van der Waals roots, to become the most general possible quadrupole interaction. The full scope of this theory becomes apparent when the orientational pseudo-potential, in the mean-field approximation, is re-parametrized, allowing access to portions of the parameter space never previously studied. The use of the mean-field approximation ensures that the symmetry of the interaction is reflected in the resulting theory, at the cost of correctness of the critical behavior and resulting in a theory which is phenomenological. The phase diagram appears to have two distinct regions where different types of uniaxial phases, those with the long axes aligned (rod-like), and those with short axes aligned (plate-like) are stable. These regions are generally separated by a region where isotropic, plate-like, rod-like, and biaxial phases meet. Some theories imply a duality 2 transformation relating rod-like and disk-like phases . In such theories, the duality transformation is the identity in a region where all the phases meet. Such a region is therefore referred to as a self-dual point. 3 Theoretical Model Inter-particle Interaction For this analysis, a particle is considered to be a superposition of three mutually-orthogonal axes. Each axis has some quantifiable characteristic which determines the strength of its interaction with the other axes. This may be thought of as the "length" of an equivalent vector, although it has no geometric significance, and in fact this "length" need not be a scalar. The energy of interaction between two arbitrarily-oriented unit vectors n° and n6 is written Uot=f(la,lt,r).[(na.nt)2-i], [1] where f(la,lt) is some function of the characteristics L and l t of the vectors and r is the distance between the particles. The energy must be symmetric with respect to interchange of the two vectors, therefore the function f is constrained to be symmetric with respect to its arguments. The part of [1] in square brackets is the complete orientation dependence for this interaction, and has a number of important features. Firstly, it reflects the axial nature of the vectors (they represent axes), in that the energy of interaction is unchanged if either or both of the n's is replaced by -n. Secondly, the expression as written does not depend on the reference frame in which the computation is performed. Finally, the average of the orientation-dependence over all possible orientations of (say) n° with respect to n l is zero. Consider now two particles, each consisting of three mutually-orthogonal vectors interacting as above. Define a£„=i(3-nS-nJ-^„) [2] where n* is the -^component (x, y, or z) of the k"1 unit vector (k=l, 2 or 3), in some reference frame. Then the quantity 4 Uoi=-|-x,,.<rSV^'.h(r) [3] is the energy of interaction between particles a and b, where the orientations of the two particles are reflected in the a's. (The constant factor is for later convenience, with the sign chosen so that mutually-aligned orientations are favored.) The requirement that U0j be symmetric with respect to particle interchange forces the matrix xi;- to be s ymmetric. This matrix completely characterizes the interparticle interaction, as can be readily seen on examination of the terms of the right-hand side of [3] for which the x, ; are coefficients. They have the form (summing over a and /?) (a)i (b)j / (<>)«' _ ( » ) > \ 2 1 m * O / J -"pa = (n n ) - I , [4j clearly equivalent to equation [1], sharing the important properties of invariance under reflection of any of the axes, reference-frame invariance, and zero-averaging. Equation [3] can be separated into a sum of orientation-dependent terms of type [4], each term being multiplied by some coefficient x,; which characterizes the interaction but is orientation-independent. Making contact with the work of Bergersen et al^, this orientation-independent part is written xo-=iU.-CrKiC,-) [5] which is manifestly symmetric. Each vector now has two characteristics, which may be thought of as an "inducer" of the interaction and a "response", as they are in the Van Der Waals interaction of Bergersen et al^, so that the full interaction between two vectors (say a and b) is now the "induction" of a, £a, multiplied by the "response" of b (or "susceptability"), Ct> plus the "response" of ^ a to the "induction" of b, Ca£t. This, however, is merely an intuitive guide to the nature of these coefficients - the theory as written makes no assumptions about the physical origin of the interparticle interaction. Terms may be added to [5] in such a way that the orientational interaction is unaffected. In particular, terms of the form 5 do not modify the interparticle interaction energy [3]. The second term, mutliplied by the orientation-dependent part [4] (summing over i and j , of course) yields zero, j independent of a. In fact, taking y{j to be any symmetric matrix, it is possible to determine a. Rewrite the diagonal terms of [6] as ^JjUg) ( i = l j 2, 3) [7] with the (t nonzero. This can then be substituted into each of the off-diagonal equations of [6], followed by multiplication of each of the resulting equations (i, j) by CiCj, yielding i(y.Ta)-c/+i(y,Ta)-c.-2+(a-y.-i)c,-cj = o. (i*j) • [8] Since the matrix y is symmetric, there are three such quadratic equations, which can be formally solved to yield the set of results Ci=ki*C2> C2=k2"C3> Ci=k3-C3 [9] where the k's are the quadratic root formula for the appropriate equation from [8]. These k's are functions of a only, and the requirement for a consistent solution to [9], namely k!k2=k3, [10] is therefore an equation for a. Thus, any symmetric matrix ytj can be written in the form [6], and the expression [5] for the orientation-independent part of the interparticle interaction energy is completely general. Note that the £'s and C's need not be real since the roots k1} k2 and k3 may in general be complex, but the pseudopotential is always real. Mean Field Theory 6 An n-particle system obeying such a two-particle interaction has total energy <« =- H ix,., ( ^ • • E h ^ j ) ^ . + .S ' . f : h ( r 2 1 ) ^ + -)}, [ii] or, f .^Ix^-Q^Qjo [12] where Q = (cr), and the angle brackets denote an average over all the particles in the system. Assuming that each particle is acted on only by an average field, and averaging over separations so that h(r)=l, the single-particle pseudopotential is written e, = -| :x j j{crL'J ,'Qj 0-iQ' / 5Qj 0}, [13] where the second term ensures that = et. This is the potential energy, in the mean-field approximation, of a particle with orientation given by a in a fluid of particles whose average orientation is Q. As all particles experience the same interaction, the subscript 1 is superfluous, and will henceforth be dropped. The partition function can now be written Z=^|dnexp ( - /?0 , [14] where dQ = sin(0)d0-dci-dV>, with 0, ci, and V being the usual Euler angles specifying the orientation of the particle, following the convention of Goldstein^. The factor l/8?r2 ensures that Z=l in the isotropic case Q=0. Before continuing with the thermodynamic aspects of the model, it is useful to define parameters analogous to those of Bergersen et al^ as follows: 7 where £= | ( c ; i+c : 2 -K3) C=|(G+C2+C3)- The convention is that £ 3 > c ; 2 > c ; i - this fixes the order of the axes, so that the same constraint cannot be simultaneously applied to-the C's. The u's are clearly biaxiality parameters, and the x's (not to be confused with the matrix x from above) quantify the longitudinal anisotropy of the particle interaction. Inverting equations [15] gives ^1=ix1(3u1-l)+|, ^ - i x ^ U j + l H I , £ 3=Xi -K , Ci=|x2(3u2-1)+C, C2=4x2(3u2+1)+C, C3=x2+C-[16] which allows the pseudopotential [13] to be written entirely in terms of the parameters [15]. With the matrix x having the form [5], this will be a sum of terms of the type ti*' = i ^ M a 1 - , 2 ) + a 3 ) [17] independent of £, where the a and /? subscripts on the <r-matrices have been dropped. Terms of the type £jQ', Ci-a', a n d C.Q' a re constructed along the lines of [17] in the obvious way. Once again following Bergersen et al^, note that in the reference frame in which the average is diagonal, Q'-Q2= i(D-C) -IP+c) D Q2 -i(S-P) -J(S+P) [18a] and 1 2_ a -a — -i(d-c) -J(d+c) 3 a = -i(»-p) •J(s+P) [18b] where 8 s=i(3cos20-l), p=-§(sm20cos2cS), d=-f(sin20cos2vO, C=§(l+cos20)cos2<£cos2 -^3cos0sin2cisin2 ,^ [19] and S=(s>, P=(p), D=(d>, C=(c>. The off-diagonal terms in [18b] are nonzero in general, but do not contribute to the orientational potential, which is the trace of the matrix product, and have therefore been omitted. The last four quantities in [19] are the order parameters of Bergersen et al^, 8 each proportional to one of the order parameters used by Straley . Substituting [17] and [18] into [13], and performing the summation over a and /?, yields the single-particle orientational pseudopotential parameterized analogously to that of Bergersen et al^: c=-ix1x2[|(S+u1D)(s+u2d)+f(S+u2D)(s+u1d) +i(P+u1C)(p+u2c)+i(P+u2C)(p+u1c) [20] -f(S+u1D)(S-fu2D)-|(P-r-u1C)(P-fu2C)]. Note that this pseudo-potential, and therefore the theory, is invariant with respect to interchange of U j and u2. The free energy can now be written /?F=-ln[^ydQ exp(-/?0 [21] Differentiating this with respect to the order-parameters S, D, P and C, and setting the resulting expression to zero, gives ^ F ) _ n _ lT_A a s = 0 =- J^ydn(-^[3s+|(u1-ru2)d-3S-i(u1-|-u2)D].exp(-/?e) or 3S+§(ui+u2)D =(3s+§(u1+u2)d). [22] 9 Linear combinations of [22] and the similar equations resulting from differentiation of [21] with respect to D, P and C, result in the expected self-consistent equations, S=(s), P=(p), D=(d), and C=(c) [23] where now the angle brackets denote an ensemble-average, <x) = i/|%-exp(-/?£) for a single particle, rather than (equivalently) the particle-average for a single system. It is convenient for later analysis to introduce a new parametrization, which reflects the equivalence of U j and u2, specifically u=(u1-(-u2), v=uru2, [24] 2 where u and v take on any values. Particularly, when v > ux and u2 are complex conjugates, whose sum and product (and therefore, the orientational pseudopotential) are real. This region is not available to a simple u l5u2 parametrization. In terms of this parametrization, the pseudopotential [20] becomes c=-ix1x2[2(2S+uD)s + f(uS+2vD)d + i(2P+uC) + |(uP+2vC) . -f(S2+uSD+vD2) - i(P2+uPC+vC2)]. 1 J The Order Parameters Each of the four order parameters describes an independent type of ordering for a biaxial (or in some cases, also for a uniaxial) particle. This is evident if the definitions of the parameters [19] are studied. The S order-parameter is the usual uniaxial nematic order parameter , and does not depend on the biaxiality of the particles. It quantifies the tendency for a particle to 10 line up its long axis with the z-axis of the principle ordering frame. The P order parameter is also independent of the biaxiality of particles, although it reflects a biaxial type of order. It measures the tendency of a particle to have a <j> Euler-angle near 0 or ir when 6 is near 7r/2, that is, the tendency of a particle to lean in the x (or -x) direction, if it leans at all. In the Goldstein^ convention, the <f> rotation is the second rotation, that of the z' axis about the z axis. This is a biaxial order parameter. The D order parameter is similar to the P order parameter, but involves the V rotation (of the body about the z' axis). It measures the tendency of a particle to align its x-axis with the principle ordering frame z-axis when the z-axis is not aligned, that is, when sin20 is large. D is therefore a uniaxial order parameter. The C order parameter is the most complicated, but in a sense the most intuitive of the four. It measures the tendency of a particle to line its axes up with those of the principle ordering frame in general — it is the "obvious" biaxial order parameter. Special Cases Certain portions of the u,v parameter space correspond to models studied previously in the literature. In particular, those studied by Freiser3 and Straley2 are particularly relevant. Freiser^ points out that his inter-particle orientational interaction has, in a cartesian representation, the form U = -Trace(R-1QRQ) [26] where U is the interaction energy, Q is the quasi-quadrupole tensor, and R is the (unitary) operator representing the rotation which takes the orientation of one particle into the other. This provides an immediate parallel with [3]. In the principal axis frame of a particle, aa0 — (|&ia-\)bap- [27] 11 In the cartesian representation, the rotation operator is the matrix whose columns are the components of the (normalized) basis vectors of the rotated frame, written in the unrotated frame. Equation [2] is recovered by the application of the inverse rotation ' operator on the left, and the rotation operator on the right, of [27], (K'ha^Rps = naynT-(l6ia-l)Sa? = fn'7nH^. [28] The quantity §£,cr' , in equation [3] with x , - therefore , corresponds to Freiser's Q tensor. Freiser's theory^ has implying Ui=u2, and is consequently less general than the full theory. The order parameter Q0 is proportional to S-fUiD, and the biaxial order parameter Q2 is proportional to the biaxial order parameter P + U i C . This model has been studied extensively in the literature, most notably by Remler and Haymet . The degree of biaxiality r of Remler and Haymet is 2j(3/2)u = J(3/2)ui, and the "self-dual" condition is met when u=|. o The correspondence between this model and that of Straley is very straightforward. Stralejr's inter-particle interaction is V = a + /?F, + 7[F2+F3] + 6F4, [29] where F 1 = s, F2=-|p, F3=-§d, and F4=|c. [30] The parameter a does not enter into the mean field computation which Straley performs. The interparticle interaction can be cast into this form, by taking one particle to be in its principal axis frame, and another to be rotated by appropriate functions s, d, p, c. The interparticle interaction is y=-l(iiCi+^CiHP4o [31] as above, which can be written, after some algebra, 12 V=-|F1+lu(F2+F3)-?vF4. [32] The procedure, then, is to take the Straley excluded-volume parameters, L, B, and W, and compute the quantity /? according to Straley's equation (9). Then, scale L, B, and W by a common factor so that /?=-§, and use the new scaled L, B, and W to compute the Straley parameters 7 and 6. Comparison of equations [29] and [32] gives that u=§7, and v=-|6. This procedure generally gives parameters such that U j ^ u 2 , but both u's are real and positive. The "self-dual" condition here is satisfied when, for instance, L=10, B = ^ [lO, W=l, corresponding to u=0.8447 and v=0.0943. 13 Analysis Convergence The analysis of the general theory presented above is .carried out by numerical iteration of the self-consistent equations [25]. Consider a simplified version of [22], where the partition function is Z=y0exp(/?Ss :iS2). ' [33] The free energy can then be immediately written /?F=-ln[Z0]+i/?S2 = -In leading to the self-consistency equation |0exp(/?Ss) +|/?S2, [34] V - ° = - i f + m If F is minimized, then or /?(<s2>-<s>2) < 1. [37] If the self-consistencey equation [35] is iterated with a value of the order parameter S which is of the form Seq+6, where Seq minimizes the free energy, the new order parameter from the equation will be, to first order in 6, b n e w " /dfi.exp(/?Ss)(l-f-/?*s) • l 3 S J Multiplying top and bottom by Z'0l gives 14 <s> -I- /?6<s2> ° n e w ~ 1+/J6<s> and, expanding the denominator in powers of 6 and keeping only first-order terms, [39] Snew = S e q + 6(/?[<s2>-<s>2]). [40] The coefficient multiplying 6 is of course equation [37], and is between zero and one if the free-energy is a minimum at Seq- Thus, in the case that some S minimizes the free energy, the ordinary intuitive iteration procedure will converge to that S, and in fact will also diverge from local maxima. The numerical analytical task, therefore, is firstly to evaluate the integrals which will allow the self-consistent equations to be iterated, and secondly to observe the resulting solutions. All computations are performed with x1=x2=l. SPIPHAS The principal tool used for this is a program called SPIPHAS, written in FORTRAN IV and running on the UBC mainframe MTS system. Since all the self-consistent equations [23] have the same denominator, it is necessary to evaluate five different integrals — the four numerators and the denominator, which being (within a factor of Sn-2) equal to Z0, can also be used to compute the free energy once the interation sequence is judged to have converged. A variety of programs were written to accomplish the iteration task, in various approximations, but the only program which generated useful results quickly was SPIPHAS. Unlike much of the other software, which was designed with high-accuracy order parameter calculations in mind, SPIPHAS was designed primarily to find the high-temperature and low-temperature boundaries of the uniaxial phases, those with nonzero S and D, but zero P and C. The basic assumption is that P and C are small — the pseudopotential in the integrand can then be Taylor-expanded, thus removing the <j> dependence from the argument of the exponential. Logically, the structure of SPIPHAS is very simple. The outermost routine, NEWOPS, takes old "guess" order parameters, and runs them through the self-consistent equations in the small-P, small-C approximation once, returning the next 15 iteration. A value for the free-energy is also returned, computed using the new order parameter estimates and the partition function, called ZDEN. The outer loop of the NEWOPS routine is a simple Simpson's Rule integration, J over V>, on the five functions mentioned above plus a sixth function used for diagnostic purposes. This sixth function can be given an arbitrarily pathological form to give a -bound on the error in the outermost numerical integration, and to assist in choosing an optimal number of points to integrate over. This outer loop is, in fact, the only numerical integration in SPIPHAS, which fact is largely responsible for the relative efficiency and usefulness of this program over its more exacting counterparts. Each function evaluation of the outer loop calls the ININT routine to perform the other integrations. ININT embodies the small-P, small-C approximation. The expansion of the exponential of -^dependent terms is taken to fourth order, with second-order phase transitions at low temperatures in mind. The program is not expected to give accurate values for P and C as (say) functions of temperature, but merely to correctly locate the phase boundary between the biaxial and uniaxial phase. The myriad variables POCO, P0C01, and so forth, are oS-integrated terms of the expansion, including factors of s and d for the numerators for those respective order parameters. These terms have been integrated manually before being inserted into the program, and are all simply powers of cos{<j>) and sin(c>). The resulting expressions are functions of rp, which is treated as a constant within ININT and is integrated over numerically in NEWOPS, and of 0, or rather //=cos(0). Since fx appears only as p in the exponent, the //-integration can be performed analytically in terms of error functions. In the case where coefficient of y} is positive, it is necessary to change to imaginary variables and compute a complex error function, whose result is nevertheless real. Further, since only even powers of n appear as coefficients (The function c could provide an odd power, but is eliminated by the <j>-integration), it is possible to express all the //-integrations by decomposing the integrals by parts successively, and evaluating only the final, //"-integral using the error function technique. 16 Support Software Of course, SPIPHAS, which simply iterates the self-consistent equations once, is not a stand-alone numerical-analtyical tool. There are three separate interface modules, or "front ends" which operate with SPIPHAS. The most primitive of these is in some sense also the most powerful. Called "FR0NT4", it is an interactive interface which allows a user to specify starting order parameters, temperature, and anisotropy. This allows one to play with the self-consistent equations, and possibly develop some kind of feel for what kind of solutions can be expected under what circumstances. It is, furthermore, a valuable diagnostic aid in the event that the SPIPHAS program begins misbehaving under one of the other interfaces. The user, using FR0NT4, can single-step through procedures and decide if the anomalous behavior is in the physics or in the software. The second interface, AUT04, is most useful for generating lists of order parameters as a function of temperature. AUT04 prompts for a starting and ending temperature, and a number of steps, and a starting guess for the parameters S, D, P and C, and of course values for u and v, and then simply drives the equations through until they converge. For each step in temperature, AUT04 uses as a starting guess the converged solution from the previous temperature. Far from transitions, this greatly enhances the convergence rate, since the new solution is near the old one. Of course, this does not hold when a transition is passed, where solutions may cease to exist, or move rapidly in (S,P,D,C)-space. Finally, the MAP interface is used for mapping the phase diagram. Given a u and v coordinate, MAP uses SPIPHAS to find the lowest-free-energy solution it can, and tracks solutions upward in temperature, noting where, for instance, a biaxial solution becomes uniaxial (heating through a second-order transition), or where a uniaxial solution becomes metastable as its free energy exceeds that of the isotropic phase, which is zero. These two types of transitions are the only kind that MAP looks for. First, MAP finds a biaxial solution at low temperature, either rod-like (starting guess with S>0) or plate-like (starting guess has S<0), and tracks the one with the lowest free-energy upward in temperature until the biaxial order parameters P and C 17 are smaller than the basic noise-level. These .parameters are then set to zero, and the resulting uniaxial solution is tracked upward until it's free energy becomes positive. Thus, as a function of u and v, MAP can identify the transition temperatures at the upper and lower end of the uniaxial phase (if any), and determine which kind of uniaxial phase (rod-like or plate-like) is stable. All of the SPIPHAS drivers share an important element, namely the convergence-criterion. It is up to the front end to decide whether or not a solution has converged, and all of the front ends use the criterion that the sum of the squares of the differences between the new and the old parameters should not exceed 10"9. In the worst case, a single parameter differing from the next step by 10"4'5 could be called "converged". It is for this reason that order parameter values less than 10~4 are considered equivalent to zero. Results The order parameter S was determined as a function of temperature in the simplest case, that of u=v=0, to check that the result was consistent with the Maier-Saupe5 result. Figure 1 shows the temperature dependence of the S order parameter in the uniaxial phase, with the temperature in absolute energy units. (/?=1/T). The discontinuity at the right hand end of the curve is the first-order phase transition at T=0.33, with the correct5 value of S at the transition of 0.43. In this solution, the other order parameters D, P, and C are all zero. 12 The phase diagram of Remler and Haymet was investigated by choosing parameters such that U!=u2, or v=u2/4. In particular, the case u=0.3266, corresponding to Remler and Haymet's degree of biaxiality r=0.2 was mapped. Figure 2 shows the curves for the order parameters C (near-vertical on the left) and S as a function of o temperature, with u=0.3266, v=0.0266. As has been pointed out by Straley among others, the order parameters are coupled in pairs, S with D and P with C. Therefore, C measures the presence of the biaxial phase, and S that of the uniaxial phase. This figure shows a spectacularly rapid second-order transition at T=0.07 from the biaxial to the uniaxial, and a first-order uniaxial to isotropic transition at T=0.34, with a discontinuity in S of 0.4. The ratio of first-order transition temperatures between this case and the Maier-Saupe of approximately 1 agrees with the mean-field results quoted 18 in Table I of Remler and Haymet6, although the discontinuity in S in this case is not consistent with that table. The observed ratio of the temperatures of the transitions, uniaxial-to-biaxial of 4.86 (0.35/0.07) corresponds reasonably well with the ratio indicated by Figure 1 of Remler and Haymet . Figure 3 shows a spectacular example of the behavior of the two phases in the "self-dual" case, u=2/3, v=l/9. The upper curve is the C order parameter and the lower curve is S. Since this was computed in a small-C, small-P approximation, the temperature behavior of the C order parameter below T=0.3 should not be taken too seriously, although of course in a perfectly ordered phase C could be as large as 3. The figure shows a second-order transition at T=0.4 directly from the biaxial phase to the uniaxial phase. o Next, the phase behavior of the model in the region investigated by Straley was examined. In figure 4, the self-dual case L=10, B=4l0, W=l corresponding to u=0.8447, v=0.0943, is illustrated. Again the parameters are C and S, with C being the upper curve. At the transition, it appears first-order, as if there were a discontinuity in both order parameters, masked by the line-drawing routine which draws the graphs. However, inadequate resolution on the temperature scale cannot be ruled out as the cause of this strange appearance. Figure 5 shows a classic rod-like biaxial through uniaxial to isotropic transition sequence, this one with L=10, B=2.5, W=l corresponding to u=0.65730, v=0.0506. This is in the region that Straley identified as exhibiting this sequence, and the ratio of transition temperatures of about 3 is approximately matched on Straley's phase diagram. Figure 6 is the plate-like region, with a negative S parameter. Here, L=10, B=3.5, W=l giving u=0.9297, v=0.1204. As indicated by Straley, the plate-like phase is stable in this region. At this point it is instructive to consult figure 7, which is a map of the u,v plane with certain landmarks indicated. The x marks the location along the parabola Ui=u2 12 where the Remler and Haymet self-dual solution is located. Towards the origin on this parabola lie rod-like phases, and away from it lie plate-like phases. Point R is the rod-like phase of Straley, point D is his self-dual phase, and point P is a plate-like phase. This suggests the possibility of a separatrix of some kind joining D and X, and possibly other points on the plane. Row 11 of Table I shows that the stable phase in the region u=0.5, v=0.2 is plate-like. It seems plausible that there will be a "self-dual" solution somewhere between u=0.5,v=0.2 and the rod-like area of the parabola directly 19 below it. Table I shows the output of the mapping program in this region, along the line u=0.5 from v=0.1 to v=0.2. Negative numbers in the T ^ column do not indicate negative temperatures, but rather that the stable uniaxial phase is plate-like in this region, rather than rod-like. The point marked A on figure 7 is the approximate location of the expected cusp. However, Table I has the perplexing result that for some values of v, the biaxial-uniaxial transition occurs ABOVE the uniaxial-isotropic one. Of course, this is not physical — this may indicate a first-order transition to the isotropic phase from the biaxial, as was suggested by the self-dual temperature-dependence at point D. It is possible that there is a widening separatix radiating from the point X, although this result at D is contradicted by Straley's pha se diagram2. 20 Conclusions The model described is a generalization of several other biaxial liquid crystal models. Although based on the work of Bergersen et al^, the resulting model contains as subsets several other well-studied liquid crystal models, most notably those of Freiser^ and Straley . The formulation as presented here admits any symmetric matrix xi;-, and is therefore the most general theory constructable from the quadrupolar invariants. There remains some ambiguity with regard to the behavior of the phase diagram in the region off of the parabola U!=u2, but the observed behavior is not inconsistent with an earlier studied model. The behavior in the newly-accessible region v > u2/4, above the parabola, is qualitatively similar to the behavior below. There is suggestive evidence that there is a line or zone of isotropic to biaxial transitions radiating from the Freiser self-dual point, which separates plate-like uniaxial phases from rod-like uniaxial phases. Figure 1. Order parameter S as a function of temperature i n the Maier Saupe model. 22 Figure 2. Order parameter C ( l e f t curve) and S (right curve) vs temperature for u=0.3266, v=0.0266. 2 3 Figure 3 . Order parameter C (upper curve) and S (lower curve) i n the case u=l / 3 , v= 1 / 9 . o Figure 4. S (lower curve) and C (upper curve) vs. T for case u=0.8447, v=0.0948, the s e l f dual case i n Straley's model. in 2 5 Figure 5. S (right curve) and C ( l e f t curve) for u=0.6573, v=0.0506, rod-like case i n Straley's model. .6 26 Figure 6. S (lower curve) and C (upper curve) vs T for u=0.9297, v=0.1204, p l a t e - l i k e case i n Straley's model. 0.0 0.063 0.125 0.188 0.25 TEMPERATURE 2 7 F i g u r e 7 . Phase diagram map. m o 28 Table 1: MAPPER output for u=0.5, 0.1 < v < .2 u V Tnb TNI 0.5 0.1 0.2625 0.375 0.5 0.11 0.3125 0.375 0.5 0.12 0*3125 0.375 0.5 0.13 0.3125 0.375 0.5 0.14 0.3625 0.375 0.5 0.15 0.3625 0.375 0.5 0.16 0.4125 0.375 0.5 0.17 -.4125 0.375 0.5 0.18 -.4125 0.375 0.5 0.19 -.3625 0.350 0.5 0.20 -.3625 0. 375 29 Bibliography j 1. A. Saupe, P. Boonbrahm and L.J. Yu, J . de Chimie Physique, 80, 3 (1983). 2. J . Malthete, L. Liebert, A.M. Levelut and Y. Galerne, CR. Acad. Sc. Paris t303, Serie II, 1073 (1986). 3. S. Chandrasekhar, B.K. Sadashiva, B.R. Ratna and V.N. Raja, Paramana - J . Phys., 5, L491 (1988). 4. B. Bergersen, P. Palffy-Muhoray and D.A. Dunmur, Liquid Cryst., 3, 347 (1988). 5. B.G. Wu, B. Ziemnicka and J. W. Doane, J . Chem Phys, 8 8 , 1373 (1988). 6. B.O. Myrvold and P. Klaeboe, Spectrochemica Acta, 42A, 1035 (1986). 7. S.R. Sharma, P. P a l f f y Muhoray, B. Bergersen and D.A. Dunmur, 32, 3752 (1985). 8. J.P. Straley, Phys. Rev A10, 1881 (1974). 9. H.-Goldstein C l a s s i c a l Mechanics, 2nd Ed. Addison Vesley (1980). 10. M.J. Freiser, Mol. Cryst. L i q . Cryst. 14, 165 (1971). 11. M.J. Stephens and J.P. Straley, Rev. Mod. Phys. 46, 617 (1974). 12. D.K. Remler and A.D.J. Haymet, J . Phys. Chem, 90, 5426 (1986). 30 C C THIS PROGRAM TAKES AS INPUT THE STRAYLEY SHAPE PARAMETERS C L,B,W WITH L>B>W, AND, AFTER SCALING FOR SIZE, OUTPUTS C THE CORRESPONDING U1,U2 SHAPE PARAMETERS FOR OUR THEORY. C ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c REAL*8 L,B,W,S,U1A,U1B,U2A,U2B,A,B,C,D,BETA,DELTA,GAMMA C 5 CALL FWRITE(6,'Enter L,B,W.: ') CALL FREAD(5,'R,R,R: ',L,B,W) IF (L*B*W.EQ.0.0) GOTO 10 C C COMPUTE STRAYLEY'S BETA, THEN SCALE L,B,W SO THAT C BETA COMES OUT TO -3/2. C BETA=(-2.*B*(W*W+L*L)-2.*W*(L*L+B*B)+L*(W*W+B*B)+8.*W*B*L)/3. CALL FWRITE(6,•BETA=<R>.: 1,BETA) S=(3./(2.*DABS(BETA)))**(1./3.)*(-DSIGN(1.DO,BETA)) CALL FWRITE(6,1S=<R>.: »,S) L=S*L B=S*B W=S*W BETA=(-2.*B*(W*W+L*L)-2.*W*(L*L+B*B)+L*(W*W+B*B)+8.*W*B*L)/3. CALL FWRITE(6,•BETA=<R>.: ',BETA) C C NOW COMPUTE Ul, U2. C GAMMA=0.5*(L*L-B*W)*(B-W) DELTA=-L*((W-B)**2) C A=(-9./8.) B=GAMMA 3 1 C = 0 . 2 5 * D E L T A D = ( B * B - 4 * A * C ) C A L L F W R I T E ( 6 , • C O E F S A , B , C = < R > , < R > , < R > . : • f A , B , C ) C A L L F W R I T E ( 6 , 1 D I S C R I M I N A N T = < R > . : ' , D) C U 1 A = ( - B + D S Q R T ( D ) ) / ( 2 * A ) C U 2 A = ( - B - D S Q R T ( D ) ) / ( 2 * A ) C 3 0 C A L L F W R I T E ( 6 , • U l , U 2 = < R > , < R > . : ' , U 1 A , U 2 A ) U = U 1 A + U 2 A V = U 1 A * U 2 A C C A L L F W R I T E ( 6 , ' U , V = < R > , < R > . : • , U , V ) C U = G A M M A *(8./9.) V = D E L T A * ( - 2 . / 9 . ) C C A L L F W R I T E ( 6 , ' U , V = < R > , < R > . : • , U , V ) G O T O 5 1 0 S T O P E N D 32 C C THIS IS A FRONT-END FOR THE 4-ORDER-PARAMETER C DELTA NOT EQUAL TO DELTA STUFF. C MOSTLY JUST DEMANDS STUFF, NOT TOO HELPFUL. C ccccccccccccccccccccccccccccccccccccccccccccccccccc c REAL*8 OS,OD,OP,OC REAL*8 S,D,P,C,F,T,U1,U2 C LOGICAL*l CMD(4),OLD(4)/ ,O l, ,L*, lD 1, 1 '/ LOGICAL*l NEW(4)/,N',,E'f,W',1 1/,STP(4)/ 1S 1,'T•, 1O','P'/ LOGICAL*4 EQCMP C S=0.5 D=0.1 P=0. 0 C=0.0 C CALL FWRITC(6,1CC=ON;1) CALL FREADC(5,'UC=ON;') C 17 CALL FWRITE(6&Temp and an i s o t r i p i e s , T, Ul, U2: ') CALL FREAD(5,'R*8,R*8,R*8: ',T,U1,U2) C 10 CALL FREAD(5,'S: ',CMD,4) IF (EQCMP(4,CMD,STP)) GOTO 20 IF (EQCMP(4,CMD,OLD)) GOTO 15 IF (.NOT.(EQCMP(4,CMD,NEW))) GOTO 17 CALL FWRITE(6,1&NEW OPS, SDPC.: ') CALL FREAD(5,1R*8,R*8,R*8,R*8: ',OS,OD,OP,OC) GOTO 21 33 15 OS=S OD=D OP=P OC=C 21 CALL FWRITC(6,'CC=OFF;') CALL NEW0PS(0S/0P,0D,0C,T,U1/U2,S,P/D,*C/F) CALL FWRITE(6,•S,D,P,C=<R*8>, <R*8>, <R*8>, <R*8>.: ' , S , D , V , C ) CALL FWRITE(6,• F=<R*8>.: • fF) CALL FWRITC(6,•CC=ON;•) GOTO 10 20 CALL FWRITE(6,* I t i s a far, far better thing etc. etc.: ') STOP END 34 C C THIS PROGRAM IS TO PHAS4 AS NEWPHAS IS TO MUPHAS. C C IT COMPUTES A NEW SET OF ORDER PARAMETERS, C BASED ON AN INPUT OLD SET, S,P,C,D, BY ITERATING C THE SELF-CONSISTENT EQUATIONS IN THE SMALL-P, C SMALL-C APPROXIMATION. C C THIS PROGRAM HAS THE SAME HIGHEST-LEVEL INTERFACE C AS APHAS, SO THAT THIS AND APHAS CAN SHARE FRONT ENDS. C C LANDAU COEF VERSION IS SPIPH2, A WHOLE NEW PROGRAM. C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC c SUBROUTINE NEWOPS(S,P,D,C,TMP,UP1,UP2,NS,NP,ND,NC,F) C REAL*8 S,P,D,C,TMP,UP1,UP2,NS,NP,ND,NC,F C REAL* 8 PSI,PSIMAX,PSIMIN,DPSI,PMULT,CFACT,PI REAL*8 SNUM,PNUM,CNUM,DNUM,ZDEN,SO,DO,PO,CO,Z0,XNUM INTEGER*4 NPTS/71//N,I C PI=4.DO*DATAN(1.DO) C C PERFORMS AN N-POINT SIMPSONS INTEGRATION. C N=NPTS-1 PSIMAX=2.0*PI PSIMIN=0.0 DPSI=(PSIMAX-PSIMIN)/N C PSI=PSIMIN 3 5 C A L L I N I N T ( T M P , U P 1 , U P 2 , P S I > S,D,P,C,SO,DO,PO,CO,ZO) SNUM=SO DNUM=DO PNUM=PO CNUM=CO ZDEN=ZO C X N U M = P S I P S I = P S I + D P S I C PMULT=4.DO DO 1 0 0 1 = 2 , N C A L L I N I N T ( T M P , U P 1 , U P 2 , P S I , S , D , P , C , SO, DO, PO, CO., ZO) SNUM=SNUM+SO*PMULT DNUM=DNUM+DO*PMULT PNUM=PNUM+PO*PMULT CNUM=CNUM+CO*PMULT ZDEN=ZDEN+ZO*PMULT C XNUM=XNTJM+PS I * PMULT P S I = P S I + D P S I P M U L T - 6 . 0 - P M U L T 1 0 0 C O N T I N U E C C A L L I N I N T ( T M P , U P 1 , U P 2 , P S I , S , D , P , C , S O , D O , P O , C O , Z O ) SNUM=SNUM+SO DNUM= DNUM+ DO PNUM=PNUM+PO CNUM=CNUM+CO ZDEN=ZDEN+ZO XNUM=XNUM+PSI C C L A S T B U T NOT L E A S T , C L E A N 1 EM U P A B I T . C 3 6 C F A C T = D P S I / 3 . D 0 S N U M = S N U M * C F A C T D N U M = D N U M * C F A C T P N U M = P N U M * C F A C T C N U M = C N U M * C F A C T Z D E N = Z D E N * C F A C T X N U M = X N U M * C F A C T C C A L L F W R I T E ( 6 C H E C K I N T E G R A L = < R > . : 1 , X N U M ) C C O K , O K , S O I ' L L M A K E I T E F F I C I E N T L A T E R , G I M M E A B R E A K ! C N S = S N U M / Z D E N N D = D N U M / Z D E N N P = P N U M / Z D E N N C = C N U M / Z D E N C F = 0 . D 0 F = - D L O G ( Z D E N / ( 8 * P I * P I ) ) + 1 0 . 2 5 * ( 1 / T M P ) * ( 3 . D O * ( N S * N S + U P 1 * N S * N D + U P 2 * N D * N D ) + 2 ( N P * N P + U P 1 * N P * N C + U P 2 * N C * N C ) ) C R E T U R N E N D C C F O L L O W I N G , T H E I N I N T R O U T I N E . T H I S R O U T I N E I S N O T C S T A G G E R I N G L Y C O M P L E X , B U T I T I S S T A G G E R I N G . C I T H I N K F O R S H E E R L E N G T H , I T M I G H T W I N T H E P R I Z E . C C O U L D B E B R O K E N U P , B U T I T W O U L D N ' T B E A N Y M O R E C R E A D A B L E . C ccccccccccccccccccccccccccccccccccccccccccccccccccccc c S U B R O U T I N E I N I N T ( T , U , V , P S I , S , D , P , C , S O , D O , P O , C O , Z O ) c R E A L * 8 T , U , V , P S I , S , D , P , C , S O , D O , P O , C O , Z O 3 7 C R E A L * 8 B E T A , O E 1 , O E 2 , O E 3 , O E 4 , U B A R , A , K l , K 2 , A K 1 , A A , P I R E A L * 8 I 5 , I 4 , I 3 , I 2 , I 1 , I 0 , C 2 P S I , S 2 P S I C O M P L E X * 1 6 A R G , R E S COMMON / D I A G / O E l , O E 2 C C T H E S E V A R I A B L E S A R E T H E S U M C O M P O N E N T S C R E A L * 8 P O C O , P 0 C 0 1 , S P O C O , D P O C O , P 0 C 2 , P 0 C 2 1 , S P 0 C 2 , D P 0 C 2 R E A L * 8 P 0 C 4 , P 0 C 4 1 , S P 0 C 4 , D P 0 C 4 , P 1 C 1 , P 1 C 1 1 , S P 1 C 1 , D P 1 C 1 R E A L * 8 P 1 C 3 , P 1 C 3 1 , S P 1 C 3 , D P 1 C 3 , P 2 C 0 , P 2 C 0 1 , S P 2 C 0 , D P 2 C 0 R E A L * 8 P 2 C 2 , P 2 C 2 1 , S P 2 C 2 , D P 2 C 2 , P 3 C 1 , P 3 C 1 1 , S P 3 C 1 , D P 3 C 1 R E A L * 8 P 4 C 0 , P 4 C 0 1 , S P 4 C 0 , D P 4 C 0 R E A L * 8 A l , A 2 , B I , B 2 , B 3 , C I , C 2 , D l , D 2 C C F I R S T , A S S E M B L E T H E P A R T S O F T H E P S E U D O P O T E N T I A L . C P I = 4 . 0 * D A T A N ( 1 . D 0 ) C C 2 P S I = D C O S ( 2 . * P S I ) S 2 P S I = D S I N ( 2 . * P S I ) U B A R = 0 . 5 * U O E l = 3 . D O * ( S + U B A R * D ) O E 2 = 3 . D 0 * ( V * D + U B A R * S ) O E 3 = P + U B A R * C O E 4 = ( V * C + U B A R * P ) B E T A = ( 1 / T ) A = 0 . 5 * B E T A K l = ( 3 . / 2 . ) * ( O E l + O E 2 * C 2 P S I ) K 2 = ( l . / 2 . ) * ( O E l + 3 . D 0 * O E 2 * C 2 P S I ) C C WHEW! O K , NOW DO T H E M U - I N T E G R A T I O N S . C A K 1 = A * K 1 3 8 A A = D E X P ( A * ( K 1 - K 2 ) ) C I F ( A K 1 . L T . 0 . 0 ) G O T O 2 0 A R G = D C M P L X ( 0 . D O , D S Q R T ( A K 1 ) ) C A L L C E R F ( A R G , R E S ) I 0 = D S Q R T ( P I / A K 1 ) * D I M A G ( R E S ) * D E X P ( - A * K 2 ) G O T O 3 0 2 0 C O N T I N U E I 0 = D S Q R T ( P I / D A B S ( A K 1 ) ) * D E R F ( D S Q R T ( D A B S ( A K 1 ) ) ) * D E X P ( - A * K 2 ) C 3 0 I 1 = ( 1 / A K 1 ) * ( A A - 0 . 5 * I 0 ) 1 2 = ( 1 / A K 1 ) * ( A A - ( 3 . / 2 . ) * I 1 ) I 3 = ( 1 / A K 1 ) * ( A A - ( 5 . / 2 • ) * I 2 ) I 4 = ( 1 / A K 1 ) * ( A A - ( 7 . / 2 - ) * I 3 ) I 5 = ( 1 / A K 1 ) * ( A A - ( 9 . / 2 . ) * I 4 ) C C O K , NOW T H E R E A L L Y W E I R D S T U F F . F I L L I N A L L T H E C T Y P E S O F T E R M S N E E D E D F O R A L L T H E S E R I E S . C I T ' S N O T T H E C L E A R E S T WAY O F D O I N G T H I N G S , B U T I T C M A K E S S E N S E I F Y O U R E A D T H E N O T E S T H A T GO W I T H I T . C C T H E V A R I A B L E S P X C Y A R E T H E I N T E G R A L S O V E R P H I A N D M U C O F T H E E X P O N E N T I A L T I M E S ( P * * X ) * ( C * * Y ) . T H E V B L S C S P X C Y , D P X C Y A D D A N E X T R A F A C T O R S A N D D , R E S P E C T I V E L Y , C T O T H E I N T E G R A N D . C P 0 C 0 = 2 . * P I * I 0 P 0 C 0 1 = 2 . * P I * I 1 C S P 0 C 0 = 0 . 5 * ( 3 . 0 * P 0 C 0 1 - P 0 C 0 ) D P 0 C 0 = ( 3 . / 2 . ) * C 2 P S I * ( P 0 C 0 1 - P 0 C 0 ) C C A l = ( 9 . / 4 . ) * P I * C 2 P S I * C 2 P S I 3 9 A 2 = 9 . * P I * ( 1 - 0 . 5 * C 2 P S I * C 2 P S I ) C P 0 C 2 = A 1 * I 2 + A 2 * I 1 + A 1 * I 0 P 0 C 2 1 = A 1 * I 3 + A 2 * I 2 + A 1 * I 1 ' C S P 0 C 2 = 0 . 5 * ( 3 . 0 * P 0 C 2 1 - P 0 C 2 ) D P 0 C 2 = ( 3 . / 2 . ) * C 2 P S I * ( P 0 C 2 1 - P 0 C 2 ) C C B l = ( 8 1 . / 1 6 . ) * ( 3 . * P I / 4 . ) * ( C 2 P S I * * 4 ) B 2 = ( 8 1 . * 3 . / 4 . ) * P I * ( S 2 P S I * * 4 ) B 3 = ( 8 1 . * 3 . * P I / 8 . ) * ( ( S 2 P S I * C 2 P S I ) * * 2 ) C P 0 C 4 = B 1 * I 4 + ( 4 . * B 1 + B 3 ) * I 3 + ( 6 . * B l + B 2 + 2 . * B 3 ) * I 2 + ( 4 . * B 1 + B 3 ) * I 1 + B 1 * I 0 P 0 C 4 1 = B l * I 5 + ( 4 . * B 1 + B 3 ) * I 4 + ( 6 . * B l + B 2 + 2 . * B 3 ) * I 3 + ( 4 . * B 1 + B 3 ) * I 2 + B 1 * I C S P 0 C 4 = 0 . 5 * ( 3 . 0 * P 0 C 4 1 - P 0 C 4 ) D P 0 C 4 = ( 3 . / 2 . ) * C 2 P S I * ( P 0 C 4 1 - P 0 C 4 ) C C P 1 C 1 = ( 9 . * P I / 4 . ) * C 2 P S I * ( 1 2 - 1 0 ) P 1 C 1 1 = ( 9 . * P I / 4 . ) * C 2 P S I * ( 1 3 - 1 1 ) C S P 1 C 1 = 0 . 5 * ( 3 . 0 * P 1 C 1 1 - P 1 C 1 ) D P 1 C 1 = ( 3 . / 2 • ) * C 2 P S I * ( P 1 C 1 1 - P 1 C 1 ) C C C l = ( 8 1 . / 1 6 . ) * ( 3 . * P I / 4 . ) * ( C 2 P S I * * 3 ) C 2 = ( 8 1 . * 3 . * P I / 1 6 . ) * S 2 P S I * S 2 P S I * C 2 P S I C P 1 C 3 = C 1 * I 4 + ( 2 . * C 1 + C 2 ) * I 3 - ( 2 . * C 1 + C 2 ) * I 1 - C 1 * I 0 P 1 C 3 1 = C 1 * I 5 + ( 2 . * C 1 + C 2 ) * I 4 - ( 2 . * C 1 + C 2 ) * I 2 - C 1 * I 1 C S P 1 C 3 = 0 . 5 * ( 3 . 0 * P 1 C 3 1 - P 1 C 3 ) 4 0 D P 1 C 3 = ( 3 . / 2 • ) * C 2 P S I * ( P 1 C 3 1 - P 1 C 3 ) C C P 2 C 0 = ( 9 . * P I / 4 . ) * ( 1 2 - 2 . * I 1 + I 0 ) P 2 C 0 1 = ( 9 . * P I / 4 . ) * ( I 3 - 2 . * I 2 + I 1 ) C S P 2 C 0 = 0 . 5 * ( 3 . 0 * P 2 C 0 1 - P 2 C 0 ) D P 2 C 0 = ( 3 . / 2 . ) * C 2 P S I * ( P 2 C 0 1 - P 2 C 0 ) C C D l = ( 8 1 . / 1 6 . ) * ( 3 . * P I / 4 . ) * ( C 2 P S I * C 2 P S I ) D 2 = ( 8 1 . * P I / 1 6 . ) * S 2 P S I * S 2 P S I C P 2 C 2 = D l * I 4 + D 2 * I 3 - 2 . * ( D 1 + D 2 ) * I 2 + D 2 * I 1 + D 1 * I 0 P 2 C 2 1 = D l * I 5 + D 2 * I 4 - 2 . * ( D 1 + D 2 ) * I 3 + D 2 * I 2 + D 1 * I 1 C S P 2 C 2 = 0 . 5 * ( 3 . 0 * P 2 C 2 1 - P 2 C 2 ) D P 2 C 2 = ( 3 . / 2 . ) * C 2 P S I * ( P 2 C 2 1 - P 2 C 2 ) C C P 3 C 1 = ( 8 1 . * 3 . * P I / 6 4 . ) * C 2 P S I * ( I 4 - 2 . * I 3 + 2 . * I 1 - I 0 ) P 3 C l l = ( 8 1 . * 3 . * P I / 6 4 . ) * C 2 P S I * ( I 5 - 2 . * I 4 + 2 . * I 2 - I 1 ) C S P 3 C 1 = 0 . 5 * ( 3 . 0 * P 3 C 1 1 - P 3 C 1 ) D P 3 C 1 = ( 3 . / 2 • ) * C 2 P S I * ( P 3 C 1 1 - P 3 C 1 ) C C P 4 C 0 = ( 8 1 . * 3 . * P I / 6 4 . ) * ( 1 4 - 4 . * I 3 + 6 . * I 2 - 4 . * I 1 + I 0 ) P 4 C 0 1 = ( 8 1 . * 3 . * P I / 6 4 . ) * ( I 5 - 4 . * I 4 + 6 . * I 3 - 4 . * I 2 + I 1 ) C S P 4 C 0 = 0 . 5 * ( 3 . 0 * P 4 C 0 1 - P 4 C 0 ) D P 4 C 0 = ( 3 . / 2 • ) * C 2 P S I * ( P 4 C 0 1 - P 4 C 0 ) C 4 1 C D O U B L E WHEW! T H A T ' S A L L O F T H E M . E V E R Y S I N G L E B L O O D Y C I T E M I N A L L T H E S E R I E S I S M A D E U P O F L I N E A R C O M B I N A T I O N S C O F T H E A B O V E T W E N T Y - S E V E N C O M P O N E N T S . I T ' S J U S T A M A T T E R C O F C O M B I N I N G T H E M I N T H E R I G H T W A Y . Y O U R E A L L Y S H O U L D j C R E A D T H E N O T E S T H A T GO W I T H T H I S , OR Y O U ' L L N E V E R G E T I T . C C O K , I T H I N K W E ' R E R E A D Y T O B U I L D T H E O U T P U T N O W . C T H E R E ' S R E A L L Y N O T H I N G T O I T ! C Z O = P 0 C 0 + 1 ( A * A ) * ( 0 . 5 * O E 4 * O E 4 * P O C 2 + O E 3 * O E 4 * P 1 C 1 + 0 . 5 * O E 3 * O E 3 * P 2 C 0 ) + 2 ( A * * 4 ) * ( 1 . / 2 4 . ) * ( ( O E 4 * * 4 ) * P 0 C 4 + 4 . * ( O E 4 * * 3 ) * 0 E 3 * P 1 C 3 + 3 6 . * ( O E 4 * * 2 ) * ( O E 3 * * 2 ) * P 2 C 2 + 4 4 . * O E 4 * ( O E 3 * * 3 ) * P 3 C 1 + ( O E 3 * * 4 ) * P 4 C 0 ) C S O = S P 0 C 0 + 1 ( A * A ) * ( 0 . 5 * O E 4 * O E 4 * S P 0 C 2 + O E 3 * O E 4 * S P l C l + 0 . 5 * O E 3 * O E 3 * S P 2 C 0 ) + 2 ( A * * 4 ) * ( 1 . / 2 4 . ) * ( ( O E 4 * * 4 ) * S P 0 C 4 + 4 . * ( O E 4 * * 3 ) * O E 3 * S P 1 C 3 + 3 6 . * ( O E 4 * * 2 ) * ( O E 3 * * 2 ) * S P 2 C 2 + 4 4 . * O E 4 * ( O E 3 * * 3 ) * S P 3 C 1 + ( O E 3 * * 4 ) * S P 4 C 0 ) C D O = D P 0 C 0 + 1 ( A * A ) * ( 0 . 5 * O E 4 * O E 4 * D P 0 C 2 + O E 3 * O E 4 * D P l C l + 0 . 5 * O E 3 * O E 3 * D P 2 C 0 ) + 2 ( A * * 4 ) * ( 1 . / 2 4 . ) * ( ( O E 4 * * 4 ) * D P 0 C 4 + 4 . * ( O E 4 * * 3 ) * O E 3 * D P 1 C 3 + 3 6 . * ( O E 4 * * 2 ) * ( O E 3 * * 2 ) * D P 2 C 2 + 4 4 . * O E 4 * ( O E 3 * * 3 ) * D P 3 C 1 + ( 0 E 3 * * 4 ) * D P 4 C 0 ) C P O = A * ( O E 4 * P 1 C 1 + O E 3 * P 2 C O ) + 1 ( A * * 3 ) * ( 1 . / 6 . ) * ( ( O E 4 * * 3 ) * P l C 3 + 3 . * ( O E 4 * * 2 ) * O E 3 * P 2 C 2 + 2 3 . * ( O E 3 * * 2 ) * O E 4 * P 3 C l + ( O E 3 * * 3 ) * P 4 C 0 ) C C O = A * ( O E 4 * P O C 2 + O E 3 * P 1 C 1 ) + 1 ( A * * 3 ) * ( 1 . / 6 . ) * ( ( O E 4 * * 3 ) * P 0 C 4 + 3 . * ( O E 4 * * 2 ) * 0 E 3 * P 1 C 3 + 2 3 . * ( O E 3 * * 2 ) * O E 4 * P 2 C 2 + ( O E 3 * * 3 ) * P 3 C 1 ) C A N D T H E R E Y O U H A V E I T , S P O R T S F A N S ! WHAT C O U L D B E E A S I E R ? C L I K E I S A I D B E F O R E , O N E D A Y T H I S W I L L R E T U R N E N O U G H C T O M A K E L A N D A U - C O E F F I C I E N T S . B U T N O T T O D A Y . R E T U R N E N D 43 C C THIS IS AN AUTOMATIC FRONT-END FOR THE PHAS4, DDPHAS C AND SPIPHAS PROGRAMS, EACH OF WHICH COMPUTE A NEW C SET OF FOUR ORDER PARAMETERS, ETC. ETC. C IT WRITES ITS OUTPUT TO FIVE DIFFERENT UNITS. EACH C UNIT OF OUTPUT CONSISTS OF ONE OR MORE PAIRS OF VALUES, C FIRST TEMPERATURE, THEN ORDER PARAMETERS S,D,P,C, THEN C THE FREE-ENERGY F RESPECTIVELY FOR UNITS 13,14,15,16 AND 17. C C THE CURRENT DUNDERHEAD VERSION REQUIRES THAT THE UNITS BE C ASSIGNED IN THE RUN COMMAND. C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC c REAL*8 S,D,P,C,F,OS,OD,OP,OC,Ul,U2,T,TMIN,TMAX,DT REAL*8 DIST,EPS/1.D-10/ INTEGER*4 NT,ICOUNT,IMAX,I REAL*8 OEl,OE2 COMMON /DIAG/ OEl,OE2 C CALL FWRITC(6,1CC=ON;1) 10 CALL FWRITE(6,1&AUT04:: Tmin, Tmax, number of points: ') CALL FREAD(5,'R*8,R*8,I: ',TMIN,TMAX,NT) IF ((TMIN.GE.TMAX).OR.(NT.LE.l)) GOTO 10 CALL FWRITE(6,'&AUT04:: Anisotropics u l , u2: ') CALL FREAD(5,'R*8,R*8: •,U1,U2) C CALL FWRITE(6,'&AUT04:: Starting OPs, SDPC: ') CALL FREAD(5,1R*8,R*8,R*8,R*8: 1,OS,OD,OP,OC) C CALL FWRITC(6,'CC=OFF;') CALL FWRITE(6,1Running.. . : •) C 44 DT=(TMAX-TMIN)/DFLOAT(NT-1) C C FOR EACH TEMPERATURE, LOOP UNTIL OP'S CONVERGE. C T=TMIN DO 20 1=1,NT 15 CALL NEWOPS(OS,OP,OD,OC,T,U1,U2,S,P,D,C,F) DIST=(S-OS)**2+(D-OD)**2+(P-OP)**2+(C-OC)**2 IF (DIST.LT.EPS) GOTO 17 OS=S OD=D OP=P OC=C GOTO 15 C 17 WRITE(13,100) T,S WRITE(14,100) T,D WRITE(15,100) T,P WRITE(16,100) T,C WRITE(17,100) T,F CALL FWRITE(6,'AUT04::- Iteration <I> complete.: ',1) CALL FWRITE(6,'AUT04:: Computed OE1, OE2 are <R*8>, <R*8>.: 1 OEl,OE2) T=T+DT 20 CONTINUE STOP 100 FORMAT(2(IX,E20.14) ) END 4 5 C C T H E A C T U A L P H A S E - D I A G R A M M A P P E R . F I N D S B I A X I A L - T O - U N I C A N D U N I - T O - I S O T R O P I C U S I N G A N A P P R O P R I A T E N E W O P S •> C R O U T I N E . C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C c R E A L * 8 U , V , V S C L / 0 . 1 / I N T E G E R * 4 J , N V / 1 1 / C U = 0 . 5 DO 2 0 J = 1 , N V U = 0 . 5 V = 0 . l + D F L O A T ( J - l ) * V S C L / D F L O A T ( N V - l ) C A L L M A P P E R ( U , V ) C A L L F W R I T E ( 6 , ' M a p p e r : : < I > . : ' , J ) 2 0 C O N T I N U E S T O P E N D C C T H E M A P P E R P A R T , W H I C H D E T E R M I N E S T H E TWO T R A N S I T I O N C T E M P E R A T U R E S . C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C c S U B R O U T I N E M A P P E R ( U , V ) c R E A L * 8 T , U , V , O S , O D , O P , O C , S , D , P , C , F E , T U B , T N I R E A L * 8 I S P , I D P , I P P , I C P , I S R , I D R , I P R , I C R R E A L * 8 E P S , D T / 0 . 0 5 / , D I S T , N D T I N T E G E R * 4 B S T A B C COMMON / T O L / E P S 4 6 C E P S = l . D - 9 C C L O O K F O R A R O D - L I K E S O L U T I O N , A T LOW T , F I N D F R E E - E N E R G Y . C T = 0 . 1 0 I S R = 0 . 9 I D R = 0 . 1 I P R = 0 . 1 I C R = 1 . 0 C 3 0 C A L L C O N V R G ( I S R , I P R , I D R , I C R , T , U , V , F E ) F R O D = F E C C F I N D T H E P L A T E - L I K E S O L U T I O N , A N D G E T I T ' S F R E E E N E R G Y C I S P = - 0 . 3 I D P = - 0 . 3 I P P = 0 . 1 I C P = - 0 . 6 C C A L L C O N V R G ( I S P , I P P , I D P , I C P , T , U , V , F E ) F P L A T = F E C C O K . D E C I D E W H I C H O N E T O F O L L O W U P O N . C I F ( F P L A T . L T . F R O D ) G O T O 5 0 C T S I G = + 1 . 0 O S = I S R O P = I P R O D = I D R O C = I C R G O T O 6 0 4 7 C 5 0 T S I G = - 1 . 0 O S = I S P O P = I P P O D = I D P O C = I C P C 6 0 C O N T I N U E N D T = D T C C WE H A V E NOW G O T , I N O S , O P , O D , O C , T H E S T A B L E R O F T H E C R O D OR P L A T E S O L U T I O N S . I T E R A T E I N T A N D W A T C H F O R T H E C B I A X I A L O P ' S T O GO AWAY C 6 5 I F ( ( D A B S ( O P ) . L T . ( 1 . E - 4 ) ) . A N D . ( D A B S ( O C ) . L T . ( 1 . E - 4 ) ) ) G O T O 7 0 T = T + D T C A L L C O N V R G ( O S , O P , O D , O C , T , U , V , F E ) G O T O 6 5 7 0 N D T = N D T / 2 . I F ( N D T . L T . l . E - 2 ) G O T O 7 2 T = T - N D T C A L L C O N V R G ( O S , O P , O D , O C , T , U , V , F E ) I F ( ( D A B S ( O P ) . L T . ( l . E - 4 ) ) . A N D . ( D A B S ( O C ) . L T . ( l . E - 4 ) ) ) G O T O 7 0 7 1 N D T = N D T / 2 I F ( N D T . L T . l . E - 2 ) G O T O 7 2 T = T + N D T C A L L C O N V R G ( O S , O P , O D , O C , T , U , V , F E ) I F ( ( D A B S ( O P ) . G T . ( l . E - 4 ) ) . O R . ( D A B S ( O C ) . L T . ( l . E - 4 ) ) ) G O T O 7 1 G O T O 7 0 C 7 2 T U B = T * T S I G O P = 0 . 0 O C = 0 . 0 N D T = D T 4 8 C C NOW F I N D T H E I S O T R O P I C T R A N S I T I O N C 7 5 I F ( F E . G T . ( - 1 . D - 4 ) ) G O T O 7 6 T = T + D T C A L L C O N V R G ( O S , O P , O D , O C , T , U , V , F E ) G O T O 7 5 7 6 N D T = N D T / 2 . I F ( N D T . L T . l . E - 2 ) G O T O 8 0 T = T - N D T C A L L C O N V R G ( O S , O P , O D , O C , T , U , V , F E ) I F ( F E . G T . ( - 1 . D - 4 ) ) G O T O 7 6 7 7 N D T = N D T / 2 . I F ( N D T . L T . 1 . E - 2 ) G O T O 8 0 T = T + N D T C A L L C O N V R G ( O S , O P , O D , O C , T , U , V , F E ) I F ( F E . L T . ( - 1 . D - 4 ) ) G O T O 7 7 G O T O 7 6 C 8 0 T N I = T C W R I T E ( 1 3 , 1 0 0 ) U , V , T U B , T N I C R E T U R N 1 0 0 F O R M A T ( I X , 4 ( E 1 4 . 8 , I X ) ) E N D C C T H E C O N V E R G E S U B R O U T I N E — T A K E S A S O L U T I O N A N D C O N V E R G E S I T C U S I N G N E W O P S C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C S U B R O U T I N E C O N V R G ( O S , O P , O D , O C , T , U , V , F ) C 4 9 R E A L * 8 O S , O P , O D , O C , T , U , V , S , D , P , C , F , D I S T , E P S C COMMON / T O L / E P S C 1 0 C A L L N E W O P S ( O S , O P , O D , O C , T , U , V , S , P , D , C , J ) D I S T = ( S - O S ) * * 2 + ( P - O P ) * * 2 + ( D - O D ) * * 2 + ( C - O C ) * * 2 I F ( D I S T . L T . E P S ) G O T O 2 0 C O S = S O P = P OD=D O C = C G O T O 1 0 C 2 0 R E T U R N E N D
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Nematic phases in fluids of biaxial particles Reid, Andrew Charles Edmund 1989
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Title | Nematic phases in fluids of biaxial particles |
Creator |
Reid, Andrew Charles Edmund |
Publisher | University of British Columbia |
Date Issued | 1989 |
Description | The investigation of the possible uniaxial phases of a fluid of biaxial particles undertaken by Bergersen, Palffy-Muhoray and Dunmur forms the starting point for further research into the full range of possible phases of such a fluid. A generalization of their interaction model, free from constraints having to do with interaction details, retaining only the biaxial symmetry, is used in the mean-field approximation to explore the range of possible orientationally-ordered phases for such a system. This model is an equilibrium model which does not include dynamic effects. The basic inter-particle interaction is abstract, having the correct symmetry for biaxial particles, and is the most general biaxial interaction constructable from lowest-order scalar invariants. The self-consistent equations resulting from this formulation are obtained in the mean-field approximation and therefore retain both the symmetry and the generality at the cost of exact numerical correctness. Four order parameters are identified, corresponding (within a numerical factor) to those found by Straley and Freiser, as well as those of Bergersen et al. The phase diagram of a fluid of biaxial particles, then, is mapped out in terms of the behavior of these order parameters, as indicated by the self-consistent equations, as a function of three anisotropy parameters and the temperature. The primary method of analysis is iteration of the self-consistent equations obtained from differentiating the free energy. Numerical results are obtained for the location of the phase boundaries, and the temperature dependence of the order parameters in various phases. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-08-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0085073 |
URI | http://hdl.handle.net/2429/27625 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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