Improved analysis of the effect of chain flexibility on the nematic to smectic-A phase transition

Chemical Physics Letters 556 (2013) 113–116

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Improved analysis of the effect of chain flexibility on the nematic to smectic-A phase transition Prabir K. Mukherjee Department of Physics, Presidency University, 86/1 College Street, Kolkata 700 073, India

a r t i c l e

i n f o

Article history: Received 25 October 2012 In final form 22 November 2012 Available online 1 December 2012

a b s t r a c t Experimental studies have shown that the chain flexibility plays an important role in the properties of the liquid crystal phase transitions. We examine the effect of the chain flexibility on the nematic to smectic-A phase transition within the Landau phenomenological theory. We show how the flexibility of the chains influences the critical behavior of the nematic to smectic-A phase transition. The possibility of the tricritical point at the nematic to smectic-A phase transition is discussed in a phenomenological way. The theoretical predictions are found to be in good qualitative agreement with available experimental results. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction During the past 40 years, a large number of experimental studies are devoted to the nematic to smectic-A (N-SmA) phase transition. At the N-SmA phase transition the continuous translational symmetry of the nematic phase is spontaneously broken by the appearance of one-dimensional density wave in the smectic-A (SmA) phase. The experimental studies [1–4] showed that the N-SmA transition can indeed be continuous when measured to the dimensionless temperature ðT  T NA Þ=T NA  105 , where T NA is the absolute stability limit of the nematic phase. The first order N-SmA phase transition has been observed [5–7,9–11] only when the nematic range is extremely narrow. The long chain molecules of alkyl and alkanes are characteristic of their high flexibility due to their enormous variety of configurations. Thus the influence of molecular flexibility may be another factor for the first or second order character of the N-SmA phase transition. Experimental results [12,13,8] showed that a dramatic change in the behavior of the N-SmA phase transition takes due to the finite flexibility of viruses and chains. Doane et al. [12] observed the tricritical behavior of the N-SmA phase transition by changing the alkyl end chains. Th dependence of the chain flexibility on the N-SmA phase transition was studied by several authors [14–21] within density-functional theory, tube-model calculation and Monte Carlo simulations. The density-functional theory by Somoza et al. [16] and Poniewierski et al. [17,18] predicted a first-order N-SmA phase transition up to a tricritical point at a certain value of length-diameter ratio, after which the transition becomes continuous. Another density-functional study and tube-model calculation by Tkachenko [19,20] predicted the strong first order

E-mail address: [email protected] 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.11.071

N-SmA phase transition. The Monte Carlo simulation study by Polson et al. also predicted the first order character of the N-SmA phase transition. McMillan [22] introduced a specific interaction model which would predict first order as well as second order phase N-SmA transitions. In this Letter, we investigate the effect of chain flexibility on the N-SmA phase transition within Landau formalism. The purpose of the present Letter is to include the dependence of the molecular configurations and chain lengths in Landau-de Gennes model and to determine the influence of the chain flexibility on this transition. The general case of the I–N-SmA phase sequence is discussed. In this work we find a tricritical point (TCP) on the N-SmA phase transition which supports the experimental results [12]. 2. Theory The aim of this Letter is to show an overall description of the effect of the molecular flexibility on the N-SmA phase transition that can be performed of a Landau approach, in which the free energy is expressed in functions of the molecular configurations, chain flexibility and of the translational and orientational order parameters. A phenomenological theory that describes the N-SmA phase transition was proposed by de Gennes [23,24]. The layering in the SmA phase is characterized [23] by the order parameter wðrÞ ¼ w0 expðiWÞ, which is a complex scalar quantity whose modulus w0 is defined as the amplitude of a one dimensional density wave characterized by the phase W. The wave vector ri W is parallel to the director ni in the SmA phase, the layer spacing of which is given by d ¼ 2p=q0 with a nonzero q0 ¼ jrWj. The nematic order parameter proposed by de Gennes [23] is a symmetric, traceless tensor described by Q ij ¼ 2S ð3ni nj  dij Þ. The quantity S defines the strength of the nematic ordering. Let us consider a system of N chain molecules. To consider the effect of chain flexibility, we

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P define another secondary order parameter f ¼< N1 Ni¼1 fðqi Þ >, which represents the averaged extension of molecule. The value of fðqÞ is a measure of extension of chain molecule which is the ratio of the end-to-end distance in the configurations q to that in the most extended configurations. Thus fðqÞ is equal to 1 for the most extended configuration and it takes a value between 1 and 0. Further we assume only the long configurations contribute the orientational order. Thus fðqÞ ¼ 0 for short configurations. The choice of this order parameter is similar to the one defined for the phase transitions in n-alkanes and lipid membranes by Kimura and Nakano [25]. We define another quantity, r ¼ ðl  l0 Þ=l0 , which represents the variation of the lengths of the conformers. Here l0 is the equilibrium value of the length l of a molecule in the absence of order parameter. We will discuss the coupling between the primary and secondary order parameters and the quantity r for the N- SmA phase transition. Now the expansion of the free energy including the coupling of the secondary order parameter f and the quantity r with the translational order parameter w and orientational order parameter Q ij near the N-SmA phase transition can be written as

1 4 1 1 F ¼ F 0 þ aQ ij Q ij  bQ ij Q jk Q ki þ cðQ ij Q ij Þ2 þ a0 jwj2 3 9 9 2 1 0 1 2 1 1 1 2 2 þ b jwj þ pf þ mr þ cij Q ij jwj þ dQ ij Q ij jwj2 4 2 2 2 3 1 1 1 1 2 þ HQ ij Q ij r þ KQ ij Q ij f þ Gjwj f þ Ljwj2 r þ Mfr 3 3 2 2 1 1 1 2 2 þ d1 jri wj þ d2 jDwj þ eQ ij ðri wÞðrj w Þ 2 2 2

ð2:1Þ

cij ¼ cni nj and F 0 is the free energy of the isotropic phase. As usual a; a0 and p are temperature dependent parameters. Here a ¼ a0 ðT  T 1 Þ; a0 ¼ a0 ðT  T 2 Þ and p ¼ p0 ðT  T 3 Þ. T 1 ; T 2 and T 3 are virtual transition temperatures. All other parameters are assumed to be temperature independent. c; d; K; H; M; G and L are coupling constants. c is chosen negative to favor the SmA phase over the nematic phase and we choose d > 0 for the SmA phase. Physically, the coupling constants represent the change in the microscopic interactions which stabilize the nematic and SmA phases. The gradient term  e involving Q ij governs the relative direction of the layering with respect to the director. A negative value of e favors the stability of the SmA phase. The isotropic gradient terms d1 and d2 in Eq. (2.1) guarantee a finite wave vector q0 for the smectic density wave. The direct linear coupling term  jwj2 Q ij in the free energy (2.1) will be absent for describing the I-SmA phase transition since such a term cannot exist in the isotropic phase [26]. The coefficients b; c; a0 ; a0 and b are assumed to be positive. The nematic phase would be stable for a < 0, a0 > 0 and p < 0. The SmA phase would be stable for a < 0, a0 < 0 and p < 0. The isotropic phase is stable for a > 0, a0 > 0 and p > 0. Here we consider the phases in which the nematic and smectic order are spatially homogeneous, i:e. S = const. and w0 = const., and for the SmA phase a spatially constant wave vector q0 with the layering along the director. The substitution of Q ij and w in Eq. (2.1) leads to the free energy expansion 1 1 3 1 1 1 1 F ¼ F 0 þ aS2  bS þ cS4 þ a0 w20 þ b0 w40 þ pf2 2 3 4 2 4 2 1 1 1 2 2 1 2 1 2 1 2 2 þ mr þ cSw0 þ dS w0 þ HS r þ KS f þ Gw20 f 2 2 2 2 2 2 1 2 1 1 1 2 2 2 2 2 4 þ Lw0 r þ Mfr þ d1 w0 q0 þ d2 w0 q0 þ ew0 q0 S 2 2 2 2 The equilibrium value of 1

req ¼  2

ð2:2Þ

r is obtained as

HS2 þ 12 Lw20 þ Mf m

ð2:3Þ

Elimination of

r in the free energy (2.2) yields

1 1 3 1 1 1 1 F ¼ F 0 þ aS2  bS þ c1 S4 þ a0 w20 þ b1 w40 þ p1 f2 2 3 4 2 4 2 1 1 1 1 1 2 2 2 2 2 þ cSw0 þ d1 S w0 þ K 1 S f þ G1 w0 f þ d1 w20 q20 2 2 2 2 2 1 1 2 2 2 4 þ d2 w0 q0 þ ew0 q0 S 2 2 2

2

ð2:4Þ

2

H L where c1 ¼ c  2m ; b1 ¼ b0  2m , p1 ¼ p  Mm ; K 1 ¼ K  MH ; m HL d1 ¼ d  2m ; G1 ¼ G  LM . m Since the orientational order parameter decreases with chain length and the isotropic to nematic (I–N) phase transition temperature increases with chain length, (r; S) and (r; f) couplings should contribute the positive term in the free energy. So the I–N phase transition line shifts to the right side of the coordinate a in a–p plane. One also sees the shift of the N-SmA transition line. One can also expect anomalously small coefficients of the fourth order power of w0 for molecules with very ‘soft’ fragments. Thus the coefficient b1 may change from positive to negative value. Thus the NSmA phase transition can be first order of short chain lengths and second order for long chain lengths which agrees with experimental observations [12]. The dependence of the chain length on the NSmA phase transition transition was experimentally observed by Doane et al. [12]. Minimization of Eq. (2.4) with respect to S; w0 ; q0 , and f yields the following three phases:

(I) Isotropic: S ¼ 0; w0 ¼ 0; q0 ¼ 0; f ¼ 0, (II) Nematic: SN – 0; w0 ¼ 0; q0 ¼ 0; f – 0, (III) Smectic-A: SA – 0; w0 – 0; q0 – 0; f – 0.

The equilibrium value of f can be obtained as 1

feq ¼  2

K 1 S2 þ 12 G1 w20 p1

ð2:5Þ

Elimination of f in the free energy expansion (2.4) yields

1 1 3 1 1 1 1 F ¼ F 0 þ aS2  bS þ c01 S4 þ a0 w20 þ b01 w40 þ cSw20 2 3 4 2 4 2 1 0 2 2 1 1 1 2 2 2 2 2 4 þ d1 S w0 þ d1 w0 q0 þ d2 w0 q0 þ ew0 q0 S 2 2 2 2 K2

G2

ð2:6Þ

1K1 where c01 ¼ c1  2p11 ; b01 ¼ b1  2p11 , d01 ¼ d1  G2p . 1 Since the molecular conformations increase as the area per molecule increases, the coupling between ðf; SÞ and ðf; wÞ should contribute positive term in free energy. We conclude that K > 0 and G > 0. We see that for a strong enough coupling constant G; b01 can become negative. Thus if b01 < 0, Eq. (2.6) has a solution for w0 . The N-SmA phase transition is then first order. In this case sixth order term  w60 should be added in the free energy (2.6). For a weak coupling constant G, b01 remains positive. Then Eq. (2.6) has no solution for w0 , the transition is second order. Thus, when b01 < 0 we have four nonzero order parameters appearing below a second order transition, with the translational order parameter w0 is induced by the orientational order parameter S, chain flexibility r and molecular conformations f. Eq. (2.6) shows the shift of the N-SmA and I–N phase transition temperatures. What transpires from the above analysis that the molecular conformations and chain flexibility strongly influence on the character of the N-SmA phase transition. The coupling between the molecular conformations and w0 also shifts the phase N-SmA phase transition line. We will now calculate the chain length dependence of the N-SmA and I–N transition temperatures. We have shown that the N-SmA transition temperature changes with chain length. We start by using the Landau free energy expansion while taking into

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account the chain length dependence of the N-SmA and I–N transition temperatures. Then the free energy can be expressed as

1 1 3 1 1 1 1 F ¼ F 0 þ aS2  bS þ cS4 þ a0 w20 þ b0 w40 þ cSw20 2 3 4 2 4 2 1 2 2 1 1 1 2 2 2 2 2 4 þ dw0 S þ d1 w0 q0 þ d2 w0 q0 þ ew0 q0 S 2 2 2 2

ð2:7Þ

From the experimental phase diagram [12] one observe that chain length vs. temperature curve for the N-SmA, I-SmA and I–N phase transitions is not a straight line. We therefore assume the quadratic form of T 1 and T 2 ,

T 1 ðlÞ ¼ T 01 þ u1 ðl  l0 Þ=l0 þ v 1 ðl  l0 Þ2 =l0

2

ð2:8Þ

2 =l0

ð2:9Þ

T 2 ðlÞ

¼

T 02

þ u2 ðl  l0 Þ=l0 þ v 2 ðl  l0 Þ

2

where u1 ; u2 ; v 1 and v 2 are constants. We assume that the temperature and chain length regions are not small so that the nonlinear dependence of phase transitions on chain length is justified. The equations of state, describing the extrema of the free energy (2.7) are of the form

1 2 aS  bS þ cS3 þ ðc  e Þw20 þ d w20 S ¼ 0 2 a1 þ b0 w20 þ ðc  e ÞS þ d S2 ¼ 0 where a1 ¼ a0 

d21 , 4d2

e ¼

d1 e 2d2



ð2:10Þ ð2:11Þ e2

and d ¼ d  4d2 .

The conditions for the first order I–N phase transition can be obtained as

F N ðSÞ ¼ F 0 ðTÞ; F 0N ðSÞ ¼ 0; F 00N ðSÞ P 0

ð2:12Þ

From which we get 2

TðlÞ ¼ T 10 ðSÞ þ u1 ðl  l0 Þ=l0 þ v 1 ðl  l0 Þ2 =l0

ð2:13Þ

where

T 10 ðSÞ ¼ T 01 þ ðc=a0 ÞSðS  SþI—N Þ

ð2:14Þ

þ

where S ¼ b=2c. From Eq. (2.13), when S is fixed, l vs. T is not a straight line as expected. i By substituting w20 from Eq. (2.11) into Eq. (2.7), we obtain

F ¼ F 0 

ðc  e Þa1 1 1  1 S þ a S2  b S3 þ c S4 2 3 4 2b0

ð2:15Þ

We use abbreviations F 0 ¼ F 0  a21 =4b0 ; a ¼ a  d a1 =b0   ðc  e Þ2 =2b0 ; b ¼ b þ 3d ðc  e Þ=2b0 and c ¼ c  d2 =b0 . The equilibrium condition for Eq. (2.15) yields



ðc  e Þa1  þ a S  b S2 þ c S3 ¼ 0 2b0

together with Eq. (2.16) define the coordinates of the tricritical point.

3. Results and discussion In this section we compare our theoretical results on the N-SmA phase transition with the experimental results of Doane et al. [12]. Figure 1 summarizes the topology of phase diagram associated with the free energy F, the conditions (2.12) and (2.18) - (2.20) and the different values of the material parameter. Figure 1 shows a typical phase diagram using conditions (2.12) and (2.18)–(2.20). To draw the Figure 1, we chose the material parameters as follows: I–N phase transition: a0 = 0.12 J cm-3 °C1, b ¼ 0:44 J cm3 ; c ¼ 1:36 J cm3 ; T 01 ¼ 80  C, u1 ¼ 0:96, J cm3 and v 1 ¼ 3:01 102 J cm3 , I-SmA phase transition: a0 ¼ 0:12 J cm3  C1 ; a0 ¼ 0:12 J cm3  C1 ; b ¼ 0:44 J cm3 ; c ¼ 1:36 J cm3 ; T 01 ¼ 80  C, T 02 ¼ 82  C, b ¼ 2:07 J cm3 ; d1 ¼ 0:03 J cm3 ; d2 ¼ 0:7 J cm3 , u1 ¼ 1:25 J cm3 ; u2 ¼ 0:01 J cm3 ; v 1 ¼ 3:01  102 J cm3 ; v 2 ¼ 1:95  103 J cm3 ; c ¼ 0:0; e ¼ 0:1 J cm3 and d ¼ 0:82 J cm3 . The first order (a) and second order (b) N-SmA phase transitions vary only in the values of coefficients: a0 ¼ 0:012 J cm3  1 C ; c ¼ 0:016 J cm3 , d ¼ 0:02 J cm3 ; u2 ¼ 1:01 J cm3 (a) and a0 ¼ 0:012 J cm3  C1 ; c ¼ 1:11 J cm3 ; d ¼ 1:06 J cm3 ; u2 ¼ 0:0421 J cm3 (b). All other parameters were kept fixed at the values listed for the I-SmA phase transition. There are three phases: isotropic, nematic and SmA. For this system there are only three possible phase sequences: I - N, I - SmA or N - SmA. As can be seen from the Figure 1, the nematic and the SmA phase arise from the isotropic phase along the curves I–N and I-SmA or along the curve N-SmA respectively. The I–N and I-SmA phase transitions are first order transition because of the cubic invariant in the free energy expansion. The N-SmA phase transition can either be first or second order. For the first order N-SmA phase transition, the line of the N-SmA phase transition starts at the I–N-SmA triple point as shown in Figure 1. When the temperature or the chain length of the I–N and of N-SmA phase transitions coincide, a triple point appears. The region of the SmA phase shrinks and finally disappears when the I–N phase transition takes place. The coordinates of the triple point are: lTP ¼ 5 and T TP ¼ 128:5. Increasing the temperature and chain length, the first order N-SmA phase transition line changes to a second order transition at a TCP, as shown in Figure 1. The coordinates of the TCP are: lTCP ¼ 2 and T TCP ¼ 115. The

ð2:16Þ

160

ð2:17Þ

140

and the spinodal line of the coupled system is given by

F SmA ðSÞ ¼ F 0 ; F 0SmA ðSÞ ¼ 0; F 00SmA ðSÞ P 0

ð2:18Þ

o

The conditions for the first order I-SmA phase transition are given by

Isotropic

Temperature ( C)



a  2b S þ 3c S2 ¼ 0

Nematic Triple point

120 TCP

The conditions for the first order N-SmA phase transition are given by

F SmA ðSÞ ¼ F N ðSÞ; F 0SmA ðSÞ ¼ 0; F 00SmA ðSÞ P 0

ð2:19Þ

Smectic-A

100

The conditions for the second order phase N-SmA transition read

a1 þ ðc  e ÞS þ d S2 ¼ 0; F 0N ðSÞ ¼ 0; F 00SmA ðSÞ P 0

ð2:20Þ

Solving Eqs. (2.12), (2.18)–(2.20) simultaneously will determine the various phase transition lines. The tricritical point is located at the intersection of the smectic line Eq. (2.11) and the smectic spinodal Eq. (2.17). These conditions

0

2

4

6

8

Chain Length Figure 1. Possible chain length (l)–temperature (T) phase diagram in the vicinity of an I–N-SmA triple point (TP) and N-SmA tricritical point (TCP). The solid lines represent a line of first order phase transitions while the dashed line represents a second order phase transition.

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phase diagram shown in Figure 1 is very similar to that in Figure 1 of the work of Doane et al. [12] and McMillan [22]. The N-SmA tricritical point can be found by linear extrapolation of the enthalpy MHNA ðlÞ. The point l ¼ ltcp would be the truly tricritical point. For the tricritical transition, the Landau model (2.6) with sixth order term of w0 , predicts that w0  ðT tcp  TÞ1=4 , typical of a mean-field tricritical exponent. Thus the tricritical exponent b is 1 . Alternatively, for the tricritical case (c ¼ 0), the tricritical expo4 nent b, for b ¼ 0 is found to be b ¼ 14. It is well known [27] that the heat capacity behavior is in good agreement with 3D-XY predictions with T NA =T NI 6 0:93. When the nematic range becomes short, i.e. T NA =T NI P 0:93, the expected crossover to tricritical behavior is observed. For the TCP, one obtains the specific heat from the free energy 6 (2.7) (addition of sixth order term 16 dS )

 1=2 1 3=2 1=2 X Y Z 2 a0 d T tcp  T   þ  l þ  l 4 a0 a0 a0

C P ðT; lÞ ¼

ð3:1Þ

where

a0 ¼ a0 

References

d a 0 ; b0 2

X¼

a0 T 01

in qualitative agreement with the experimental results Doane et al. [12]. The tricritical behavior of the N-SmA transition has also been observed by changing the alkyl end chains [12], varying the concentration in binary mixtures [28–34] increasing the pressure [35–37] and increasing the electric field [11]. Although we have focussed on the tricritical behavior of the N-SmA transition in the Temperature - Chain length phase diagram, a similar calculation can also be applied for the Temperature - Pressure or Temperature - concentration phase diagrams. The pressure dependence or concentration dependence tricritical points can be described by the same free energy (2.7) with a ¼ aðT; ðP; xÞÞ and a0 ¼ a0 ðT; ðP; xÞÞ. In this case the chain length l should be replaced by the pressure P or concentration x in Eqs. 2.8,2.9. Then we can get the same tricritical points in the T–P or T–x phase diagrams similar to the tricritical point in the T–l phase diagram. We point out that the pressure dependence TCP was theoretically discussed by Mukherjee and Sasmal [38]. The concentration dependence TCP on the N-SmA phase transition will be discussed in a separate paper.

d d  0 a0 T 02  1 4d2 b

!

ðc  e Þ2 þ ; 2b0

d a0 u2 ; b0  d a0 v 2 : Z ¼ a0 v 1  b0

Y ¼ a0 u1 

Eq. (3.1) shows that C P of the ordered phase has a square root singularity at the TCP. We now compare our present work with the previous works [22,14–18,21,19,20] which was analyzed the same issue for the N-SmA phase transition. Firstly, our model is different from their models. The present analysis agrees with the density- functional theory and mean-field analysis [22,16–18]. However, it differs from the Flux tube calculation and Monte Carlo simulation studies. These works rules out the possibility of the TCP near the N-SmA phase transition in the chain length vs. temperature phase diagram. Whereas, our theory shows that the chain flexibility can change the order of the phase transition. The reason may be the strong coupling between the order parameters and the chain configurations (f) and chain flexibility (r). 4. Conclusion We have presented here a simple phenomenological theory analysis to describe the effect of chain flexibility on the N-SmA phase transition. The main effect of chain flexibility is to change the character of the N-SmA phase transition. The molecular configurations and chain flexibility change the Landau coefficients. The values of the Landau coefficients are changing with the change of chain length and molecular configurations. As the chain length increases, the the second order N-SmA phase transition becomes first order at the TCP. Thus we conclude that although the N-SmA phase transition is first order for a longer chain length, but the transition becomes second order for shorter chain length. Thus our results are

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