1. Introduction
Many classes of soft matter microstructures exhibit a tailored response and transport properties because of the interactions of rods1-5. Some soft matter materials are the well known Tobacco Mosaic Virus and the fd virus, the richness of these materials is due to their various ordered phases, where the most commonly are the isotropic and nematic6-8. In this work we focus on the description of the nematic isotropic phase transition in stiff and long colloidal rods from a kinetic point of view. One of the challenges of the description of the phase transition is its relation with mechanical response functions, that are the most accessible properties in an experiment. For instance the mechanical susceptibility provides us with a mechanical stability condition9. In particular it is well known that the structure factor in the limit of very small wave vectors is the isothermal compressibility, which can be probed in a light scattering experiment.
The aim of this work is to describe the orientational relaxation by monitoring the dynamic structure factor at small wavevectors in a colloid of hard, long and stiff rods initially aligned.
The idea behind it is to demonstrate what we can learn by applying the machinery of statistical thermodynamics to a simple model: hard, long and stiff rods in suspension, which exhibit the nematic isotropic phase transition. There are two motives in this approach. The former is simply that the study of a simple model provides insight, while the latter is the close relation to experiments (or computer “experiments”) showing the complex features of relevance in our model.
The nematics phase in liquid crystals has been studied using different approaches; experimentally, via dispersion of depolarized light10, by simulation7 or theoretically11-13. The pioneering Onsager’s work concerns itself with a density functional approach to compute free energy in order to describe the isotropic nematic phase transition12. Another approach is the kinetic approach of the Smoluchowski equation13, which is able to describe the nematic isotropic phase transitions. In this work we focus on the calculation of the second order parameter by using the Smoluchowski formalism in order to describe the nematic isotropic phase transition and its connection with a mechanical response function.
Because the dynamic structure factor can be monitored by means of a time resolved small angle depolarized dynamic light scattering experiment14, we follow the orientational relaxation of rods by observing the time evolution of the dynamic structure factor15. Initially the rods are perfectly aligned by using an external field, therefore they themselves return to the equilibrium state when the external field is removed. As will be seen the dynamic structure factor can be written in terms of the parameters of order two and four. The second order parameter is calculated from the equation of evolution of the orientational order parameter tensor S(τ), which is obtained from the Smoluchowski equation and the closure relation provided by Dhont16, valid at equilibrium. Like this author, we extend the validity of this closure for nonequilibrium. The fourth order parameter is obtained directly from the closure relation cited above. At the limit of the wavevector going to zero, we will show the connection of the fluctuations of the second order parameter with the self structure factor, the main quantity in the description of the nematic isotropic phase transition.
The work is organized as follows. Because the second order plays an important role we start with its definition and evaluation; therefore in Sec. 2 the Smoluchowski formalism is given, using the Maier-Saupe potential together with an appropriate closure relation, the time evolution and its equilibrium values of the second order parameter are computed. In Sec. 3 the dynamic structure factor for small wavevectors is provided; using results from the previous section, this quantity is expressed in terms of the second order parameter only. In Sec. 4 fluctuations of the second order parameter are defined. Here also is shown the relationship between them with the dynamic structure factor. Using the statistical mechanics tools the interpretation of the self structure factor as a mechanical response is also given. In Sec. 5 results are provided. We begin with general results using only experimental conditions and the closure relation over the description of the nematic isotropic phase transition, that is, as this process can be monitored by the dynamic structure factor measured in a depolarized light scattering experiment in VH geometry which considers the polarization direction of the incident light 𝑛 0 perpendicular to that of the detected scattered ligh 15, in this work the dynamic structure factor is calculated using these conditions. Connections with the mechanical response on the predictions of the isotropic nematic concentration values are given with the inclusion of the Maier-Saupe potential. The orientational relaxation is also provided for small wavevectors as function of the concentration. At the end of the section, results for the isothermal orientation susceptibility with respect to the nematic isotropic phase transition are provided. Finally in the last Section, concluding remarks are given.
2. Second order parameter
Consider a colloidal suspension formed by 𝑁 hard, long and stiff rods embedded in a solvent. Very long and thin rods are considered, with 𝐿 and 𝐷 their length and thickness respectively, whose volume fraction scales as
Here, 𝐷𝑟 is the rotational diffusion coefficient of a single non-interacting rod,
where
In equilibrium the largest eigenvalue of the orientation order parameter tensor is the well
known nematic parameter. In general the order parameters are defined as the average
of Legendre polynomials,
with the reduced time defined as
then, the Eq. (4) becomes
For homogeneous phases, S(τ) is expressed as a diagonal tensor, in case of an
isotropic phase all the components are equal to 1/3, whereas for nematics the two
small coefficients are equal and the highest is known as the nematic order
parameter. As a result the time-dependent equation for
In equilibrium, the variation of
These two solutions are positive by varying the value of
Finally in nonequilibrium Eq. (7) is solved numerically obtaining the time dependent evolution of the nematic parameter. In equilibrium, the nematic order parameter is the quantity of central importance in describing the isotropic nematic phase transition as we will see below. For nonequilibrium we will express all the properties in terms of the second order parameter, which can be written in terms of the nematic parameter, as
As in the equilibrium state, our proposal is to show the relevance of the second order parameter also in nonequilibrium states. Because of the use of the Maier-Saupe potential, Eq. (10), in order to close our model in terms only of the second order parameter it is needed a closure relation between fourth and second order parameters. The four order parameter is given by:
Where
It is necessary to express
Replacing Eq. (11) and Eq. (12) in Eq. (10), we obtain that the fourth order parameter can be expressed in terms of the second order parameter as
Thus we have the input necessary for the description of our colloid in equilibrium and during the relaxation process, that is the second order parameter.
3. Dynamic structure factor at small wavevectors
Because our main aim is describing the orientational relaxation of the colloid by monitoring the dynamic structure factor measured in a depolarized light scattering experiment in VH geometry. Initially consider rods perfectly aligned and in
Now, since, for there to be hard rod interaction,
In order to observe the orientational properties in homogeneous phases, the dynamic structure factor is monitored, as we mention it can be probed in a depolarized light scattering experiment in VH geometry. In the case of rods, the dynamic structure factor is expressed as
where the average involves a time-dependent PDF.
where self dynamic structure factor is given by
and the distinct part is written as
From Eq. (18), to compute the self and distinct part of the dynamic structure factor is
necessary to know the one body PDF
where
By choosing the directions for the polarization vectors in a convenient way20, the self structure factor is rewritten as
On the other hand, to compute the distinct part of the dynamic structure factor, we require the Fourier transform of the total correlation function
In last equations the time dependency it is also considered in the one body PDF.
The natural dimensionless units of the above expression are considering Lk=q then Dk=(D/L)q because L/D is large, then a good approximation is taking
The consequence of this aproximation is that the distinct part can be expressed in terms of the self one
Therefore, the Dynamic Structure Factor for small wavevectors of a colloidal system of stiff and long rods is given by
We note that f(q,t) is only a function of
Considering terms up to fourth power in the dimensionless wavevector in 𝑗 0 and up to the fourth order parameter contributions in the self structure factor, the averages involved in Eq. (19) are expressed
And
Therefore by the way of the dynamic structure factor and the equilibrium, the relaxation process can be studied for small wavevectors using this simple approximation. Nevertheless in the next section the relevance of the limit for wavevector going to zero is analyzed with its physical interpretation, that is the relationship between the dynamic structure factor and the dynamic fluctuations of the second order parameter.
4. Relationship between dynamic structure factor and the second order parameter dynamic fluctuations
In this section the limit of wavevector going to zero is studied, the advantage of this limit is the connection between the self dynamic structure factor and the nonequilibrium susceptibility, which is the self correlation of the second order parameter, therefore is more convenient to refer to the self part as isothermal orientation susceptibility as can be see below. One reason for the identification with only the self part is due to the fact that the susceptibility concept is related with the fluctuations of a one body microscopic property with its corresponding measurable macroscopic quantity. Here we will see that the microscopic property is the second order parameter and the macroscopic the self structure factor.
By considering the limit of the dimensionless wavevectors as going to zero, the spherical Bessel function of zero order is equal to one, therefore the dynamic structure factor can be written as
Where
Now, by using the closure relation in Eq. (30), the self dynamic structure factor can be written in terms of the second order parameter only, that is
It is important to mention that this quantity is hold for colloids in a homogeneous phases, as the isotropic and nematic, independent of any model used for the description of the colloid. In the former only the experimental condition was used whereas the second has in addition the closure relation between second and fourth order parameter.
Now in order to go into the concept of fluctuations, we start defining the nonequilibrium fluctuations of the second order parameter as
where ⟨⋯⟩ indicates a nonequilibrium average. Therefore the nonequilibrium fluctuations correlations are given by
In Eq. (32) the temporal dependency is again assumed through the
In the same way as was previously done for the dynamic structure factor, thenonequilibrium fluctuations correlations can be expressed in terms of the second order parameter only, taking into account homogeneous phases and the Maier-Saupe potential. When the closure relation Eq. (31) is used, the expression for the self nonequilibrium fluctuations of the second order parameter is exactly the same as that for the dynamic self structure factor the limit of wavevector going to zero, Eq. (31). Therefore the quantity probed in the depolarized light scattering experiment is the nonequilibrium correlations of the second order parameter in this limit and as consequence the same expressions for the self part of these properties. It is important to observe that this equivalence in the expressions is also valid when the average is in equilibrium. The relevance of this correlation is that it can be probed in a well defined experiment, in the results section the predictions of this relationship will be provided for nonequilibrium and equilibrium orientational fluctuations. We will see below the importance of the self correlation of the second order parameter in the description of the orientational relaxation and the values of the concentration at which the isotropic nematic phase transition occurs.
Now we focus on its physical interpretation. From its role played the self structure factor, with the limit of dimensionless wavevector going to zero, could be identified as an isothermal orientation susceptibility in the sense that this quantity is the second order parameter fluctuation. To observe this identification we are able to make a gedanken experiment in which an external force
where
after some usual algebraic steps9, one arrives at
where 𝜒 𝛽 is a mechanical response function due to orientation. Thus, identifying
5. Results
This section starts with nonequilibrium and equilibrium general results from the properties measured in a depolarized light scattering experiment together with those derived by the use of the colloidal model proposed in this work. Also numerical results will be analyzed, in all of them we separate our analysis according to the concentration values of the colloid in equilibrium, that is for isotropic and nematic phases. Thus for simplicity we only use and denote isotropic when we refer to the former and nematic for the latter. We continue describing the behaviour of the second order parameter together with its relaxation time and in the same manner the temporal evolution of the dynamic structure factor. The second subsection will describe the relaxation of the colloid by means of the dynamic structure factor for small wavevectors. Finally in the last subsection results for the limit of the wavevector going to zero will be described, focusing on the isothermal orientation susceptibility.
5.1 General properties
The nematic isotropic phase transition in a colloid suspension of stiff and long rods is well
known. From our colloidal model, equation Eq. (33) predicts the value of the
concentration at which this transition occurs, that is,
On the other hand, the isotropic to nematic spinodal concentration
As was already mention in this section the results are separated in two regions depending on whether the concentration of the colloid is in isotropic or nematic phase when it is in equilibrium. From our kinetic model, Eq. (33), it is only possible to move from nematic isotropic phase transition, thus with this restriction the separation will be for lower than isotropic concentration and larger than it, which we call isotropic and nematic for simplicity, as was already mentioned, that which is lower than 𝜙 𝐼 =32/9 is isotropic and if larger is nematic.
The analysis starts considering the most general expression for the self structure factor
dependent on time,
Now, by replacing Eq. (33) in Eq. (34) we find that
which is satisfied in the isotropic or in the nematic phase, depending to the concentration. We must mention that the expression given in Eq. (29) has already been reported for the equilibrium case .
Another important situation is when the
So
It is easy to see that the second derivative with respect to the dimensionless time is
negative, so the value one quarter for the second order parameter corresponds to
a maximum, as we will corroborate with the numerical results below. It is
important to mention that Eqs. (34) and (36) are independent of any colloidal
model, the only important assumptions are with respect to the homogeneity of the
phase, the experiment proposed and the closure relation between the second and
fourth order parameters. We must mention that these features are only held for
isotropic, that is for concentrations lower than
We use our colloidal model in order to find predictions for the nematic isotropic phase transition. Thus, in equilibrium using our model Eq. (8) in the general predictions, the second order parameter equal to one half corresponds to a concentration
Focusing on the nonequilibrium behaviour, during the relaxation the self structure factor has a maximum only for
5.2 Second order parameter
The second order parameter, Eq. (9), is plotted in Fig. 1 as function of time for different concentrations. Here the behaviour is observed the behaviour depending of whether the colloid is isotropic or nematic. For the former at long times it goes to zero whereas for the latter goes to values different from zero. The relaxation process predicts a second order parameter equal to one half for the nematic concentration as it was corroborated from our model in the previous subsection.
Another observation in Fig. 1 is that the second order parameter takes the value of one quarter only for concentrations in isotropic. Taking only the equilibrium values for the second order parameter as function of the concentration the results are reported in Fig. 2. We can appreciate that for cocentrations greater than or equal to 3.6 the system is in the nematic phase, since
Here we clearly see, nematics is derived from Eq. (14) whereas the isotropic region with the help of a perturbation analysis is derived, as was already mention in the previous subsection. From Fig. 2 we observe three different regions: when the concentration is lower than 𝜙 𝐼 any isochoric process has only one stable point, for concentrations between 𝜙 𝐼 and 𝜙 𝑁 will have two stable points, finally for concentrations larger than 𝜙 𝑁 has only one point stable again. In the middle region any isochoric line will have two points, one in isotropic phase and the second in nematics. In this manner this model predicts different behaviour depending of the value of the concentration.
Defining the relaxation time
5.3 Orientational relaxation for small wavevectors
The dynamic structure factor for small dimensionless wavevectors is analyzed in this section.
An experiment is assumed in which a colloid of stiff and long rods are perfectly
aligned by means of an external field; at
equation, Eq. (12), solved with the initial condition of a full alignment of the rods and using Eqs. (15), (19) and (22).
In Fig. 3 is reported results for different values for
The important feature is that for small dimensionless wavevectors all the plots have similar qualitative behaviour. Results for a concentration in nematic is not reported because these present similar characteristics, the only important difference is that they do not have a maximum but they go into a plateau according to their concentration, but the equivalent behaviour for small dimensionless wavevectors is similar, which is the important detail.
In Fig. 4 the dynamic structure factor is reported as a function of the concentration, the plot shows the evolution of the dynamic structure function for different times and two values of the dimensionless wavevector
In this Figure each symbol corresponds to a different time, for
In a previous subsection was predict that the self structure factor in the limit of dimensionless wavevector going to zero has a maximum when the second order parameter takes the value equal to one quarter, independent of the concentration, within isotropic only. Therefore from Eq. (14), this maximum corresponds to
Now, in Fig. 6 is reported results for the dynamic structure factor for the same values as Fig. 5. Curves for intermediate times looks like a continuos phase transition as in a dipolar colloid20. This behaviour is a remanent of the external field used for the initial alignment. The main information of the phase transition comes from the self dynamic structure factor at the limit of dimensionless wavevector going to zero, showing the relevance of this limit.
5.4 Isothermal orientation susceptibility
As was shown in a previous section, it is reasonable to identify the self structure factor of the dimensionless wavevector equal to zero as a mechanical susceptibility, which we called isothermal orientation susceptibility.
From Fig. 5 is observed that during the relaxation
process for times larger than 0 the isothermal orientation susceptibility takes
values lower, higher or equal to one, only at
In the isotropic and for
With respect of the relaxation process the isothermal orientation susceptibility is able to predict when the rods have an alignment to 45° with respect to the direction of the polarization direction, in this situation the isothermal orientation susceptibility has a maximum, in isotropic, depending of the value of the concentration of the colloid, time is necesary to get this maximum, in nematic the maximum of
6. Conclusions
In this work we propose a simple theoretical model that allows us to observe the relaxation process of a colloidal system formed by stiff and long rods, via the dynamic structure factor, which has as input the order parameter two. The study is based on the Smoluchowski formalism neglecting the hydrodynamic interactions together with the Maier-Saupe potential for the interaction between them.
The advantage of having an explicit expression for the dynamic structure factor lies in its possible comparison with simulation and experimental data that allow us to test our theoretical results.
At wavevector equal to zero was shown that the dynamic structure factor is the collective correlations to the fluctuations of the second order parameter. Its self part was also identified with an isothermal orientation susceptibility. The relevance of this quantity is that is able to predict the isotropic and nematic concentrations in which the colloid has the nematic isotropic phase transition through the second order parameter in equilibrium, we find that
The results were supported with the assumption of the Maier-Saupe potential for a colloid with a nematic isotropic phase transition, thus one would expect that these predictions could be valid for different colloids which hold the same conditions.