# Entropic Behavior of Binary Carbonaceous Mesophases

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 1.**The comparison of three states of the material: a perfect crystalline solid with positional and orientational order (a), a liquid crystal with positional disorder and partial orientational order (b) and an isotropic liquid with positional and orientational disorder (c).

**u**direction in Figure 2) are preferably oriented along a particular direction called the director

**n**(Figure 2). In discotic nematic liquid crystals (DNLCs) the director

**n**is perpendicular to the long axis of discotic molecules (Figure 2). In rod-like nematic liquid crystals (RNLCs)

**n**is parallel to the long axis of the molecules (Figure 2). Carbonaceous mesophases (CMs), first reported by Brooks and Taylor [5], are DNLC mixtures obtained from petroleum pitches and synthetic naphthalene precursors [6]. The composition, polydispersity, and molecular orientation of CMs play a significant role on the final properties of cokes [7], carbon foams, carbon/carbon composites [8], and carbon fibers [9,10,11,12].

**Figure 2.**Schematic of nematic liquid crystals (NLCs), the director n is the average of the molecular orientation u, and its classification into discotic and rod-like molecules. In discotic nematic liquid crystals the director n (average orientation) is perpendicular to the long axis of the molecules; however, in rod-like nematic liquid crystals the director n is parallel to the long axis of the molecules.

_{i}is the mole fraction of i

^{th}component. For uniaxial phase

**Q**is given in terms of a temperature-dependent scalar order parameter s(T) and the average molecular orientation or director

**n**: $Q=\mathrm{s}\left(nn-I/3\right)$, where

**I**is the unit tensor. For binary discotic nematogens at equilibrium we find collinear directors $\left({n}_{1}={n}_{2}\right)$ and the mixture uniaxial scalar order parameter then is:

**Figure 3.**Classification of the mixtures into two types; type A with the non-ideal behavior and type B with ideal behavior, based on their intrinsic properties: molecular weight asymmetry ΔMw and molecular interaction parameter β. For weakly interacting mixtures and /or small molecular weight asymmetry the NI transition line exhibits a minimum by increasing the concentration (region A); however, for sufficiently strongly interacting mixtures and /or sufficiently large molecular weight asymmetry the NI transition line is monotonic (region B) [1].

- (i)
- isotropic (I) : s
_{1}= 0, s_{2}= 0. - (ii)
- nematic (N
_{12}) with s_{1}≥ s_{2}. - (iii)
- nematic (N
_{21}) with s_{1}≤ s_{2}.

_{12}to N

_{21}transition within the nematic phase due to the concentration effect. For the later case s

_{1}= s

_{2}where N

_{21}converts to N

_{12}; at this point nematic mixture behaves like a single component system, and the concentration corresponding to this transition is the critical concentration m

_{1c}at which the NI transition temperature is a minimum. The results of our previous study [1] show that depending on the molecular weight difference and the molecular interaction, two types of uniaxial nematic mixtures arise (Figure 3): (i) non-ideal mixtures; both kinds of transitions are observed within non-ideal mixtures, and (ii) ideal mixtures; only the NI transition takes place within this type of mixtures; N

_{12}exists for all range of concentration; as a result, N

_{21}and the critical concentration do not appear in the ideal mixture. Ideal binary mixtures arise under sufficiently strong interaction and sufficiently high molecular weight differences, while non-ideal mixtures arise under weak interaction and small molecular weight differences (asymmetries). As each carbonaceous mesophase mixture leads to different carbon fiber structure, it is of a great importance to be able to determine the type of the mixture.

**Figure 4.**Schematic of the phase diagram of the binary mixture which includes three states: (i) isotropic (I), (ii) nematic (N

_{12}) and, (iii) nematic (N

_{21}). In N

_{12}(N

_{21}) the higher (lower) molecular weight species has a higher molecular order parameter than the lower (higher) molecular weight component: s

_{1}> s

_{2}(s

_{2}> s

_{1}). Depending on the type of the mixture N

_{21}can appear and the transition within the nematic phase between N

_{21}and N

_{12}takes place (for non-ideal mixtures) or it can disappear (for ideal mixtures). The concentration which corresponds to the transition within the nematic phase is the critical concentration [1].

_{r1(2)}is the value of the entropy for a single component system. This value is universal at the NI transition. Our objective is to find out the effect of concentration, molecular weight asymmetry and the interaction between the components on the (i) entropic values and the entropic jump at transition and (ii) the rate of entropy change which is needed to calculate the values of the specific heat.

- (i)
- To determine the entropic behavior of ideal and non-ideal mixtures.
- (ii)
- To calculate heat capacities of the two types of mixtures.
- (iii)
- To determine their entropic variation at transition.

## 2. Maier-Saupe Binary Mixture Model

_{A}, k

_{B}, T are the Avogadro’s number, Boltzmann’s constant and temperature respectively, and E

_{mix}, S

_{mix}, and Z are the internal energy , the orientational entropy, and the partition function of the mixture per molecule respectively. E

_{mix}is given by the summation of three contributions:

_{mix}as the entropy of mixing per molecule and is given by:

_{i}, ${\mathrm{\Phi}}_{\mathrm{i}}$ and m

_{i}are the partition function, the partial internal potentials, and the mole fraction (concentration) of the i

^{th}species respectively.

_{1}, s

_{2}we find:

_{1}= 0,1) correspond to the pure NLC. As discussed in our pervious work [1] for the present CM case, the mixture is uniaxial and hence the relative director aperture is α = 0. The thermodynamics of the mixture is defined by the dimensionless temperature T

_{r}and two effective mole fractions $\left({\mathsf{\phi}}_{\mathrm{i}};\mathrm{i}=1,2\right)$:

_{i}of the two components:

_{1}= 1,400 [32] and vary the lower molecular weight of the second component, Mw

_{2}, so that the molecular weight asymmetry ΔMw = Mw

_{1}-Mw

_{2}> 0 changes.

_{1},s

_{2}); the two material parameters are β and ΔMw; the thermodynamic phase diagram is obtained by sweeping over temperature T

_{r}and concentration m

_{1}. Equations (7a&b) are solved by the Newton-Raphson method, using an eighth order Simpson integration method. Stability, accuracy, and convergence were ensured using standard methods [33]. In the discussion of results we use the following nomenclature for dimensionless (reduced) entropy:

## 3. Specific Heat

_{pr}, can be obtained by the orientational entropy which results from the ordering of the mixture [34]:

_{pr}of the nematic phase and C

_{pr}of the isotropic phase. According to this equation two terms have direct contributions in the value of the specific heat: temperature, ${\mathrm{T}}_{\mathrm{r}}$, and the rate of entropy, $\left(\frac{\partial {\mathrm{S}}_{\mathrm{r}}}{\partial {\mathrm{T}}_{\mathrm{r}}}\right)$. Depending on the trend of $\left(\frac{\partial {\mathrm{S}}_{\mathrm{r}}}{\partial {\mathrm{T}}_{\mathrm{r}}}\right)$, the trend of $\mathrm{\Delta}{\mathrm{C}}_{\mathrm{p}\mathrm{r}}$ as a function of temperature changes. Once it increases monotonically, ΔC

_{pr}increases; however, once it decreases ΔC

_{pr}can either increases or decreases, depending the magnitude of each term, ${\mathrm{T}}_{\mathrm{r}}$ and $\left(\frac{\partial {\mathrm{S}}_{\mathrm{r}}}{\partial {\mathrm{T}}_{\mathrm{r}}}\right)$. In this case, ΔC

_{pr}does not behave monotonically and exhibits a local extremum.

## 4. Results and Discussion

- (i)
- NI transition temperature: ${\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{N}\mathrm{I}}/\mathrm{W}=0.22019$
- (ii)
- scalar order parameter in the N phase at transition: s
_{NI}= 0.4289 - (iii)
- Latent heat at the NI transition: $\frac{{\mathrm{W}{\mathrm{s}}_{\mathrm{N}\mathrm{I}}}^{2}}{2{\mathrm{k}}_{\mathrm{B}}{\mathrm{T}}_{\mathrm{N}\mathrm{I}}}=0.417719$

#### 4.1. Entropic Behavior

_{r}as a function of ordering s, for a pure mesogen and an schematic for a generic mixture. As expected, entropy decreases with increase in ordering. At the NI transition the pure component displays the universal values (s

_{NI}= 0.4289, ${\mathrm{S}}_{\mathrm{r}\mathrm{N}\mathrm{I}}=-0.417719$), but for the generic mixture, the Figure shows a characteristic deviation for the universal transition values, which are discussed and quantified in detail in the rest of the paper.

**Figure 5.**Schematic of the effect of concentration m

_{1}, molecular weight asymmetry ΔMw and the interaction parameter β on the type, entropy and ordering of the mixture. It shows the general trend of the dimensionless entropy S

_{r}as a function of scalar order parameter s

_{mix}. As ordering increases the entropy decreases. It also shows that diluting the pure system changes the ordering and the entropy at NI transition.

_{r}, as a function of temperature, for an ideal mixture with ΔMw = 800 and interaction β = 1 (Figure 3); each curve shows the entropic trend for a specific concentration m

_{1}. The following features are observed in this figure:

- (i)
- For any given concentration, entropy increases monotonically with increasing temperature (Figure 5).
- (ii)
- For any given temperature, entropy increases with decreasing concentration: component “1” has higher Mw, so it is more ordered in the nematic range. Therefore, as the concentration m
_{1}increases the mixture ordering increases and entropy decreases. - (iii)
- Entropy jumps at the nematic-isotropic first order phase transition that sets in at temperature T
_{NI}(the entropy of the isotropic phase is assumed to be zero; as a result, this entropy represents the orientational entropy); the entropy jump value which is defined as (entropy of the isotropic phase) – (entropy of the nematic phase) at NI transition temperature for the almost pure mixtures (m_{1}= 0.001,0.999) are essentially the universal value ${\mathrm{S}}_{\mathrm{r}\mathrm{N}\mathrm{I}}=0.417719$, and mixing just decreases the magnitude of jump. - (iv)
- Depending on the relative population of two components there are three distinct concentration regions: (i) m
_{1}$\simeq $ [0-0.1], (ii) m_{1}$\simeq $ [0.2-0.3], (iii) m_{1}$\simeq $ [0.4-1]. The rate of entropy change (δS_{r}/δT_{r}) which corresponds to the rate of ordering change in the mixture shows a rapid decrease in region (ii). Component “2” has lower Mw; as a result, it has a lower T_{NI}(eqn.10). Therefore, while component “1” tends to remain nematic (N_{12}state), component “2” tends to transform to the isotropic (I) state and lose the ordering, though due to the interaction between the components it retains a low nematic ordering. As a result, when approaching the NI transition temperature of “2” the rate of ordering change in this component decreases and because of eqn.(2) the overall rate of change in the mixture decreases. The appearance of the inflection point in the graphs corresponds to this change in the rate of entropy trend. This phenomenon becomes enhanced when the relative concentrations of both components are significant (region (ii) m_{1}$\simeq $ [0.2-0.3]).

_{1}is shown for each curve. The following features are observed in this Figure :

- (i)
- Like the ideal case increasing the temperature decreases the ordering and increases the entropy (Figure 3) of all the mixtures.
- (ii)
- Unlike the ideal case, the effect of concentration on the ordering and entropy is different at different temperatures, increasing concentration does not necessarily decreases the entropy. The following three distinctive dilution regions arise: (i) m
_{1}$\simeq $ [0-0.3], (ii) m_{1}$\simeq $ [0.4-0.6], (iii) m_{1}$\simeq $ [0.7-1]. Region (i) is located in N_{21}state (s_{2}> s_{1}) of the phase diagram (lower left region in Figure 3); in this region the lower molecular weight species “2” is the majority component. Therefore, increasing m_{1}, dilutes the mixture, decreases the ordering and increases the entropy (Figure 5). The second and the third regions, both are located in N_{12}state (s_{1}> s_{2}); therefore, the trend is opposite. As component “1” is dominant in this region, increasing its concentration makes the mixture more ordered and decreases the entropy. - (iii)
- there is a difference between region (ii) and region (iii). Like the ideal case, the rate of entropy change (δS
_{r}/δT_{r}) in the mixture shows a rapid decrease in region (ii). When approaching the NI transition temperature of “2” the rate of ordering change in this component decreases; as a result, the overall rate of change in the mixture decreases [eqn. (2)]. This phenomenon becomes enhanced in the region (ii), where m_{1}= [0.4-0.6] and the concentration of both components is significant.

**Figure 6.**Dimensionless entropy Sr as a function of dimensionless temperature Tr of ideal (with ∆Mw = 800 and β = 1) (a) and a non-ideal (with ∆Mw = 400 and β = 0.1) (b) mixtures for different concentrations, computed using eqns. (5, 12). Ideal mixtures show monotonic behavior with respect to dilution, while non-ideal mixtures show non-monotonicity due to the transition between N

_{12}and N

_{21}. The entropy behavior reflects the scalar order parameters of the species and their relative concentration.

#### 4.2. Specific Heat

- (i)
- At any given temperature, increasing the concentration decreases the ordering and increases the specific heat.
- (ii)
- By sweeping over temperature, specific heat behaves differently for different concentrations. Depending on the concentration region ((i) m
_{1}$\simeq $ [0-0.1], (ii) m_{1}$\simeq $ [0.2-0.3], (iii) m_{1}$\simeq $ [0.4-1] , (see Section 4.1) two different trends are observed in the specific heat values versus temperature: in region (i) and (iii), when the population of one of the components is insignificant, and (δS_{r}/δT_{r}) changes monotonically, specific heat increases monotonically with temperature; however, in region (ii), where m_{1}= [0.2-0.3], specific heat increases by increasing temperature and then it shows a decrease which is followed by another increase. As mentioned in section 3, the non-monotonicity of the specific heat is due to the direct contributions of the T_{r}and (δS_{r}/δT_{r}) to its value. According to section 4.1 (δS_{r}/δT_{r}) in region (ii) shows a rapid decrease in the vicinity of T_{NI}of component “2”; as a result, ΔC_{pr}decreases in this range. On the other hand, increasing temperature increases ΔC_{pr}; as a result, a minimum appears in the ΔC_{pr}as a function of temperature.

- (i)
- A transition is observed in ΔC
_{pr}trend vs. concentration. This transition takes place at the critical concentration where N_{21}→ N_{12}. Region (i) is located in N_{21}where increasing m_{1}decreases the ordering and increases ΔC_{pr}; however, regions (ii) and (iii) are both located in N_{12}where increasing m_{1}increases the ordering and decreases ΔC_{pr}. - (iii)
- Depending on the concentration region ((i) m
_{1}$\simeq $ [0-0.3], (ii) m_{1}$\simeq $ [0.4-0.6], (iii) m_{1}$\simeq $ [0.7-1], (see Section 4.1) two different trends are observed in the specific heat values versus temperature: in region (i) and (iii), when the effect of one of the components is weak, (δS_{r}/δT_{r}) changes monotonically and specific heat increases monotonically with temperature; however, in region (ii), where m_{1}= [0.4-0.6] and hence essentially no majority component, (δS_{r}/δT_{r}) changes non-monotonically (section 4.2). As (δS_{r}/δT_{r}) has a contribution to the specific heat values, ΔC_{pr}exhibits a minimum vs. temperature in this region (see section 3).

_{pr}values we investigate ΔC

_{pr}for three mixtures with different molecular weight differences (asymmetries) ΔMw = 400 (a), 600(b), and 800(c) for m

_{1}= 0.2 and β = 0.1. The results are shown in Figure 8. Based on Figure 3 all three mixtures belong to the non-ideal type. The values of the critical concentration for these mixtures are: 0.31, 0.205 and 0.11 respectively [1]. As a result, for mixtures (a) and (b) component “2” is the majority species, so they exhibit a monotonic ΔC

_{pr}; however, in mixture (c) there is no majority component and hence ΔC

_{pr}exhibits an extremum.

_{pr}values. For brevity the related graphs are not presented here.

**Figure 7.**Specific heat ΔC

_{pr}as a function of dimensionless temperature for ideal (with ∆Mw = 800 and β = 1) (a) and non-ideal (with ∆Mw = 400 and β = 0.1) (b) mixtures for different concentrations. Ideal mixtures show monotonic behavior of specific heat while non-ideal mixtures display non-monotonicity with respect to dilution. Both cases show non-monotonicity with respect to temperature whenever the relative concnetration of the components is significant.

_{pr}versus concentration is observed. This transition takes place within the nematic phase from N

_{12}to the N

_{12}state. For both ideal and non-ideal type, whenever there is no majority component in the mixture, the trend of specific heat versus temperature is non-monotonic because each component tends to show different ordering at different temperature range.

**Figure 8.**Specific heat ΔC

_{pr}as a function of dimensionless temperature for m

_{1}= 0.2 and β = 0.1 for three different molecular weight asymmetries: ∆Mw = 400 (a), 600 (b), and 800(c). All three cases are non-ideal mixtures; in case (c) the relative concentration of the components is significant, so it shows a non-monotonic trend with respect to temperature.

#### 4.3. Entropy Jump at Transitions

_{1}= 0, 1) the value of the entropy jump is the universal value 0.417 [19]. On the other hand, diluting the system disturbs the orientation and increases the entropy. As a result, there is a concentration which corresponds to the minimum entropy jump at NI transition. This concentration depends on the intrinsic properties of the mixture and is about 0.2 in the present case. Increasing dissimilarity of the components (larger ΔMw) enhances the dominancy of component ”1” and shifts the minimum into lower concentrations.

_{1}= 0 (pure system) and m

_{1c}(pseudo-pure mixture) which is located in the N

_{21}region and the second one is observed between m

_{1}= m

_{1c}(pseudo-pure mixture) and m

_{1}= 1 (pure system) which is located in the N

_{12}region.

**Figure 9.**Entropic variations at NI transition as a function of concentration for ideal (with ∆Mw = 800 and β = 1) (a) and non-ideal (with ∆Mw = 400 and β = 0.1) (b) mixtures for different concentrations. Both pure and pseudo pure mixtures have the universal entropy at NI transition. A local minimum is observed for the ideal case, where there is only N

_{12}state; however, two local minima (one in N

_{21}and the other in N

_{12}) and a maximum (in N

_{12}→N

_{21}transition, which corresponds to the critical concentration) are observed for the non-ideal case.

## 5. Conclusions

_{21}to N

_{12}is observed in this type of mixture. This transition takes place at the critical concentration and can be detected by the entropic and specific heat trend: the trend of these quantities in N

_{21}is opposite to their trend in N

_{12}. The entropic jump at the NI transition exhibits two local minima; one located in the N

_{21}region and the other in N

_{12}region. It also has a local maximum between these two regions at critical concentration, where the mixture behaves like a single component system and shows the universal entropy jump at NI transition. For both type of mixtures whenever there is a distinct majority component, ΔC

_{pr}and entropy behaves monotonically versus temperature; however, when there is no majority component ΔC

_{pr}and entropic trends are non-monotonic with temperature.

## Acknowledgements

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Golmohammadi, M.; Rey, A.D. Entropic Behavior of Binary Carbonaceous Mesophases. *Entropy* **2008**, *10*, 183-199.
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**AMA Style**

Golmohammadi M, Rey AD. Entropic Behavior of Binary Carbonaceous Mesophases. *Entropy*. 2008; 10(3):183-199.
https://doi.org/10.3390/entropy-e10030183

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Golmohammadi, Mojdeh, and Alejandro D. Rey. 2008. "Entropic Behavior of Binary Carbonaceous Mesophases" *Entropy* 10, no. 3: 183-199.
https://doi.org/10.3390/entropy-e10030183