Gerald R. Van Hecke Harvey Mudd College Clarernont. California9171I
I I
I
Therrnotropic Liquid Crystals A use of chemical potef~tial-temperaturephase diagrams
Tvoicallv most introductions to one comoonent phase eq&bria &cuss the phase behavior of substances s&h as water or sulfur in terms of the variables temperature and pressure. Our purpose here is to present liquid crystals as examoles of materials which often exhibit fascinating phase equilibria that are readily demonstrable, and discuss liquid crystalline phases not in terms of the usual pressure and temperature variables, but rather in terms of the all too often neglected alternative chemical potential and temperature variables (1). Liquid crystals fall naturally into two classes, lyotropic and thermotropic. Lyotropic liquid crystalline mesophases form on mixing of solute(s) and solvent(s) and furnish many examples of relatively simple binary and ternary phase diagrams. We shall not discuss lyotropic systems, hut rather only thermotropic liquid crystalline mesophases.' In thermotropic substances, phases arise simply from changes in temoerature and orovide manv. svstems of fascinatine" . one component phase equilibria. There are three commonly used maior classifications of thermotro~iccrvstalline mesophases: smectic, nematic, and cholesteric (the last is increasingly and preferably referred to as a twisted nematic) (2). To a rough approximation, molecules that form liquid crvstalline mesophases are rod-like. Brief structural description of liquid crystalline phases is afforded by the relative amount of positional order, the relationship between the centers of mass and orientational order, the relationshio between the principal molecular axes. Smectic mesogenerally have orientational order and two dimensional positional order, while nematic mesophases (including cholesteric as a twisted nematic) only have orientational order, with the twist of the cholesteric phase giving rise to a helical structure. Phase transitions, and the temperatures, enthalpies, and entropies thereof, are, then, reflections of the changes in structure. Phase Equilibria
Phase equilibria are conveniently described in terms of the intensive variables temperature, pressure, and chemical potential (T, P, p, respectively) where the latter, for a one component system, is just the molar Gibbs free energy. The well known criteria for phase equilibrium are T, = T#,P, =P#, d T , P ! = P~(T,P! where a and 0 denote the phases in equilibrium. The chemical potential is completely specified by only T and P for one component systems in the absence (or neglect) of external fields. Thus the chemical potential can be pictured as the surface illustrated in Figure la. The slope of the surface is defined a t any point by the two derivatives
That is, a t constant pressure the change of chemical potential with temperature is the negative of the molar entropy 3, while a t constant temperature, the change of chemical potential with pressure is equal to the molar volume V. Points of phase equilibria occur along the intersection line The term mesophase (in between phase! is t h e commonly used generic term for any liquid crystalline phase.
Figure 1. Chemical potential sure.
surfacesas a function of temperature and pres-
of two or three chemical potential surfaces, one for each equilibrium phase, as shown in Figure Ib. Customarily the projection of this intersection on the T-Pplane is used to discuss one component phase diagrams such as illustrated in Figure 2a. This is entirely reasonable since T and P are experimentally accessible directly while chemical potential is not. Occasions arise, however, when use of one of the other possible projections, p - P or p - T as shown in Figures 2b and 2c, respectively, is preferable or more illustrative of system behavior. For example, in the case of liquid crystals there are two good and practical reasons for the use of chemical potential-temperature diagrams such as the simple example in Figure 2c. First, there exist essentially no pressure-temperature data for liquid crystalline materials. Second, because of the general complexity of these liquid crystalline systems, pressure-temperature diagrams are likely to be difficult to draw. Moreover, many compounds which exhibit liquid crystalline behavior often show three, four, sometimes more, different phases (polymorphism). Pressure-temperature diagrams for such materials, while very interesting, would be quite complex, while chemical potential-temperature diagrams are still relatively simple (3). Further we show that p - T diagrams could be used to Volume 53. Number 3. March 1976 / 161
I
crystal
I
liquid
phase transitions would allow a quantitative diagram to be drawn with the assumption either that (1) the slope, that is the entropy, of each phase is constant (not unreasonable over short temperature ranges) or (2) since adaT = -S and a2plaT2 = -aS/aT = -Cp/T, slope changes could be accounted for hy taking heat capacity constant for then the entropy change is just proportional to 1/T. But, another way to estimate these diagrams which avoids the above restrictive assumptions would be t o plot the free energy function (G - H ) / T versus T . This would result in a series of straight lines, since to a good approximation the free energy function varies linearly with temperature. The only experimental data required then would be the heat capacity. Thus heat capacity data would provide sufficient information to calculate the absolute entropy or the free energy function for the crystal and thus provide minimum data required to quantify the chemical potential-temperature diagram. Unfortunately, such data are extremely scarce for liquid crystals though relatively plentiful for many molecular crystals (2,5). Thus p-T or p-P plots must, in the main, he qualitative, but not, and this is a point of emphasis, useless. Thermotropic Mesophases
Figure 2. Projections of the p, = pa phase equilibria lines on various planes.
estimate temperatures and entropies of some phase transitions occurring in liquid crystalline systems. Of course, the usefulness of chemical potential-temperature diagrams is not limited to just liquid crystals. This is a point of emphasis, for, except in very few current physical chemistry texts, 'u-T diaerams have been needlessly iznored ( 4 ) . The slopes of the w, = j q intersection line projected onto the p-P plane (constant T ) or the p-T plane (constant P ) are as drawn in Figures 2h and 2c since for most substances molar volume changes in the sequence as Vcw8t.~< VISU,d < V,., and decreases with increasing pressure a t constant temper$ure. Similarly, the molar entropies are in the order, S,,,,~I< Sllquld < Sgapand increase with increasing temperature a t constant pressure. In either a w-P or p-T plot first-order phase transitions occur a t points of intersection of these single phase lines. Moreover, just considering the w-T plot, the change in slope at an intersection point is just the change in molar entropy for the phase transition. This, in turn, is simply related-to the molar enthalpy of the phase transition by AS = A?lIT. Thus calorimetric exoeriments can define the temDerature and slope differences hut not the absolute chemical potentials. a he methods of statistical thermodynamics do in principle provide means to calculate the absolute chemical potential, but in oractice, such calculations are extremely difficult or impossible. ~ i a n t i t a t i v eestimates of p-T diagrams are oossible. however. in several ways. If an arbitrary zero of chemical potential were defined; then the calorimetric experiments determining temperature and enthalpies of the 162 / Journal of Chemical Education
As alluded to above, thermotropic liquid crystals quite often exhibit fascinating phase equilibria. While there are three basic types of liquid crystals, the phenomenon is more widespread in that there are three recognized types of nematics (including cholesteric) and a t least seven smectics (6). Moreover the interesting phase behavior of many compounds arises from the occurrence of polymorphism. Polymorphism occurs when a single compound exhibits more than one liquid crystalline phase, more than one crystalline phase, or both. Polymorphism occurs frequently in the solid state in molecular crystals, and several cases of crystalline polymorphism are discussed in terms of chemical potential temperature diagrams by McCrone (7).Crystalline polymorphism is often exhibited by materials which form liquid crystals. For example, 4-ethoxy-4'-n-nonanoyloxyazobenzene ( C ~ H ~ O C ~ H ~ N : N C ~ H ~ O C ( : O ) C ~ H I ~ ) which forms a nematic mesophase, exhibits three crystalline modifications (8). Common examples of liquid crystalline polymorphism are combinations of two smectic phases, a smectic phase and a nematic phase, or a smectic phase and a cholesteric ~ h a s eFor . any given substance, smectics exist in the lowest temperature ranges, then nematics. Not only can a compound exhibit polymorphism, the liquid crystalline mesophases, or in some cases crystal modifications, may he either enantiotropic or monotropic (9). An enantiotropic, or normal, phase transition is one between two thermodynamically, stable phases, while a monotropic one occurs on cooling a higher temperature phase to a phase which is metastable. Both enantiotropic and monotropic phase transitions may be reversible, however. In discussing the phase behavior of liquid crystals in terms of chemical potential-temperature diagrams enantiotropic systems will be mentioned first, then systems exhibiting monotropic phases will be discussed. Phase Diagrams
The simplest p-T diagram appropriate to a single enantiotropic mesophase transition is illustrated in Figure 3a where K stands for crystal, M for mesophase or, in particular, S, smectic, N, nematlc, C, cholesteric, and I for isotropl i q ~ i d On . ~ heating a t constant pressure the system will normally follow the path of lowest chemical potential. Thus the situation illustrated in Figure 3a describes crystal melts The symbol K is commonly used to denote crystalline phases to avoid confusion with the use of C for cholesteric and S for smectic liquid crystals.
Figure 4. Idealized chemical potentiaCtemperature phase diagram illustrating a suspected secandqrder smectic A to nematic trans8ions.
Figure 3. a and b, idealized chemical potential-temperature phase diagrams for a substance which exhibits polymorphism, two enantiotropic mesophases.
ing to form mesophase a t the intersection of the K and M lines, an enantiotropic phase transition. Thus M is an enantiotropic mesophase which exists as the most stable phase over the temperature range between the T K Mand TMI intersections. Above TMI, an enantiotropic transition I, the material acts as a normal liquid. temperature of M Cooling the system slowly yields exactly the opposite hehavior. Typical examples of this behavior are: p-azoxyanisole (10) (CH30-CsH4N:N(:O)C6H4OCH3)
-
117'C
137%
K t , N o I
Figure 5. Idealized chemical potential-temperature phase diagram for 5-ladc8-mmnyloxy-2-naphmoic acid showing an enantiotropic nematic mesaphase and a monotropic smectic mesophase. L09'C
L12-C
122-C
K u S - N u 1 10Z°C
103'C
K-S-I Note Figure 3a (and b) is again an idealized representation since, while the temperature intersections can be deterand the differences in slopes determined from mined AS,,.,,i,i,., no absolute slope is known. A brief comment can he made on the magnitude of the relative slooes, . . however, as generally it is observed that ASKM is large, -30 J/ mole K, while ASMI is often an order of magnitude smaller. Thus the slope differences between the K and M lines should he much larper - than the differences between the M and I lines. Simple polymorphism is illustrated in Figure 3b where two mesophases now exist, both enantiotropic. The sequence of phase transitions on heating crystal would he crystal melting to mesophase MI a t the intersection temperature, mesophase MI transforming to mesophase Mz a t the intersection temoerature. and mesoohase M?- eoine - to isotropic liquid I. ~ n . t r o pccdnges ~ for tiansitions between mesoahases are eenerallv the same order of maenitude as those for mesophase to isotropic. Thus the changes in s l o ~ e of s all lines other than the K line should he similar to each other. Examples of compounds which exhibit this type of behavior are: 4-n-butyloxy-4'-acetylazobenzenene (12) n-
-
-
CdH90-CsH,-N=NC6Hd-C(:O)CH3
cholesteryl nonanoate (13) (C27HlsOC(:O)CsH17-n) 74-C
76.3-C
K-SctC-I
92.1-C
Some ahase transitions between mesophases appear to he second-order. For example the smectic A (smktic A is one of the several smectic phases possible) to nematic transitions in homologous 4-n-alkoxybenzylidene-4'-phenylazoanilines (CnH2,+,0CsH&H:NCsH4N:NC6Hd have characteristics of second-order transitions ( 1 4 ) .Figure ~ 4 presents the p-T diagram that would describe a second-order transition for SAto N by the tangency and coalescence of the corresponding chemical-potential curves. This follows since a second-order transition will be continuous in ap/aT and no latent heat or entropy change will he observed (15). These cases have not been comaletelv exnerimentallv verified, however. The usefulness of chemical aotential-temaerature diagrams is highly manifest when discussing monotropic phase transitions. Figure 5 illustrates the hehavior of 5-
. .
-
"Considerable controversv exists as to whether or not the S. -" N transition is second-order. Our purpose here is to suggest what the chemical potential phase diagram would look like if the transition were second-order.
Volume 53. Number 3. March 1976 / 163
iodo-6-n-nonyloxy-2-naphthoic acid which exhibits smectic and nematic mesophases (16). The phase sequence which occurs on heating the crystal would he K N a t 163'C, and with continued heating N I a t 18Z°C. Cooling the N a t 18Z°C. Further cooling to isotropic liquid gives I K or the nematic 163°C gives the system a choice of N can he undercooled to 158°C. The now undercooled, metastable, nematic phase can then undergo a reversible transition to the smectic phase which is metastable with respect to the crystal. This an example of a monotropic transition and the smectic phase so formed is called monotropic. That is, this smectic phase is never the most stable one a t any temperature. If the smectic phase does not spontaneously crystallize, which it could do a t any temperature below 158"C, it could he reheated to form the nematic phase reversibly again a t 158OC. Indeed further heating of the nematic nhase so formed could continue until i t reversiblv forms the isotropic phase a t 183°C. Discussion of the kinetics of transformation of these metastahle phases is not the intent here, hut liquid crystals provide numerous examples of long-lived metastahle states that might profitably he studied from a kinetic viewpoint. Figure 6 illustrates the u-T ohase diagram for cholesteryl octanoate (17), which exhibit; smecticand cholesteric mesophases, both monotropic. Heating crystalline cholesteryl octanoate melts the material to a normal isotropic phase a t llO0C. The behavior on cooling is much more intriguing, for here, by undercooling to 9 7 T , the isotropic phase forms the monotropic cholesteric which can be further cooled to 70°C a t which point a monotropic smectic phase forms. Both of these transitions are reversible. To digress momentarily, one might wonder, with the typical interest of the synthetic chemist in clean melting points on heating, how many monotropic liquid crystalline phases have gone unreported because samples are not also observed while cooling. The complex polymorphism exhibited by ethyl+(methoxybenzilidene)-amino)-cinnamatesynthesized and reported by Vorlander as one of the first examples of polymorphism in liquid crystals (18, 19) is presented in Figure 7. This material not only exhibits enantiotropic smectic A and nematic phases hut also a monotropic smectic B and a monotropic crystalline modification. At this point the interest and usefulness of the @ - Tdiagram should be apparent. For example, no temperature has yet been reported for the KII S e transition; yet it should he observahle. An im-
-
-
Figure 6. Idealized chemical potentiaktemperature phase diagram for cholesteryl octanoate which exhibits monotropic smectic and cholesteric phases.
164 / Journal of Chemical Education
Figure 7. Idealized chemical potential-temperature phase diagram for ethyl-(p(methoxybenzi1idene)-aminokcinnamatewhich exhibits enantiotropic smectic A and nematic phases and a manotmpic smectic B and a crystal modification.
-
portant note here is that the occurrence of the missing KrI SBtemperature is predicted even by using only a qualitative diagram. Moreover there are several ways to quantify an estimate of KII to Sg transition temperature. One general way, discussed earlier, would he to start with a single known or estimated absolute entropy, that is, the slope of one line. Then the knowledge of the transition temperatures and A S values (AFfJT) would allow calculation of all other lines. In this case, however, since no temperature or I been reentropy for a phase transition involving K ~ has ported, the chemical potential line for crystal I1 would have to be known. Knowledge of the KII line and the temperaSa ture and chance in e n t r o w for Krr .. ..would then suffice in principle to completd t h e diagram. Obtaining data for the KII-SAtransition might he difficult however. With reference to Figure 6, the diagram for cholesteryl octanoate, knowledge of the slope of the crvstal chemical potential line, thetransition temperatures and AS for those transitions would allow ready prediction of the smectic to isotropic transition temperature, and the accompanying changes in ethalpy and entropy. The same orocedure could estimate all intersection temperatures shown in Figure 7, except those involving crystal 11. Transitions involving crystal I1 are the most difficult to discuss. A second method to quantify the diagram which is on better thermodynamic footing would he to redraw the diagram in terms of the free energy function. Since the free energy function can to a very cood approximation be considered linear with temperature, reasbiahle extrapolations can be made to find missing temperatures by line intersections. The free energy functions only need heat capacity data to he calculated, hut as mentioned before, there is no free energy function data and very little heat capacity data for liquid crystals. Such data are available for molecular crvstals. and free enerev -. function temperature diagrams have been used to predict transition temperatures (5).To repeat, for emphasis, even if auantitative nredictions are difficult, drawing such plots even in a qualithive manner is easy and can be an immense aid in sorting out the often complex phase equilibria present in such systems.
-
Summary
Thermotropic liquid crystals present interesting examples of one component phase equilibria which can profit-
.
ably be discussed in terms of chemical potential-temperature (rr-T) diagrams not only to aid understanding the liquid crystalline mesophases formed, but also to illustrate first- and second-order phase transitions in an interesting manner. Literature Cited (1) For introductory reading on liquid c w t a l s see: (a) B r o m . G. H., and Shaw, W. G., Chsm. Re"., 57,10d9 (19571; (h) Gray. G. W.. "MoCUI~JStructure and the Pmperties of Liquid Crystals." Academic Press. New York, 1962: (el Gray. G. W., and Winsor. P. A , -Liquid Cry8tals and Plastic crystd.... Vols. 1 and 2. John WilW & Sons. New York. 1974. (2) B r o m , G. H., Dmne. J. W., and N e 4 V. D.. "A Review ofthe Structure and Physical prnperties of ~ i w i dCrystah: Chemiesl Rubber Publishing Co.. Cleveland. Ohio, 1971. (3) For an excellent diacmian of idealized P-T, and r-P-T diagrams including aomc wperb drawings Rciamm, A,, "Phase Equilibria: AcademiePreaa, New York. 1970. Chap. 6. (4) Some mmmon text. that use chemical potential diagrams are Diekern", R. E., '"Molecular Thermodynamics," W. A. Benjamin, New Yo& 1969. Chap. 6: Daniels. F.. end Alherty, R. A,, "Physical Chemistry: 4th Ed., John Wiles & SOM. New York, 1975,Chap. I; Castellan,G. W.,"PhyricalChemistry: 2nd Ed.. Addison Wesley,Reading, Mass., Chap 12.
..
i;~~.ea, New York. 1962. (101 Kast. W.. in "Landolt-Bbmetein.'. Vol. 2, part 2% 6th Ed.. Springer, Berlin. l969, p. 266. (11) Gray, G. W., Hartley, J. B., and Jones, B., J. Chem. Soe., 1412 (19551. (12) ~ ~ ~ ~ , ~ . , ~ d S s e k m aPhyr n , HChem. . , Z (Leiplig),222.127 119631. (IS) Ennu1at.R. D.,Moi. Cryrl. Lig. Crwt., 8,247 (19691. ~ . Cvik1.B.. J o h n ~ n D.L.,andfishel. . D.L..Phys. Rev. (141 ~ o a n eJ. , W.. P a ~ k c r ,S.. A , A. 1238 (1971): McMillan, W. L.,Phyr. Re". A, 4, 1238 11971):Tores. S.,and C1adia.P. E..Phys. Re". Left., 32.1406(19741. (15) Guggenheim, E. A,. "Thcrmadynamier." 6th Ed.. American Elevier, New York, 1967,Chap. 6. (16) Gray, G. W.. and Jane.. R.. J Chem. Sor.. 236 (1955). (171 Gray, G. W., J. Chsm. Sac., 3733(19561. (16) Vorlander.D.,Z.Phya., SI.426(19301. (19) Tarr. C. E., Dennery, R. M., and Fuller, A. M.. presented at 166th Nations1 American Chemical Society Meeting. Chicsgo. August 1973.
Volume 53. Number 3, March 1976 / 165
