Monte Carlo studies of optical transmission of anisotropic suspensions

Accepted Manuscript Monte Carlo studies of optical transmission of anisotropic suspensions

N.I. Lebovka, N.V. Vygornitskii, L.A. Bulavin, L.O. Mazur, L.N. Lisetski PII: DOI: Reference:

S0167-7322(18)33997-7 doi:10.1016/j.molliq.2018.10.117 MOLLIQ 9862

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

2 August 2018 1 October 2018 21 October 2018

Please cite this article as: N.I. Lebovka, N.V. Vygornitskii, L.A. Bulavin, L.O. Mazur, L.N. Lisetski , Monte Carlo studies of optical transmission of anisotropic suspensions. Molliq (2018), doi:10.1016/j.molliq.2018.10.117

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ACCEPTED MANUSCRIPT

Monte Carlo studies of optical transmission of anisotropic suspensions N. I. Lebovkaa, N. V. Vygornitskiia, L. A. Bulavinb, L. O. Mazurb, L. N. Lisetskic a

Institute of Biocolloidal Chemistry named after F. D. Ovcharenko, NAS of Ukraine, 42

Vernadsky Prosp., 03142 Kyiv, Ukraine. b

National Taras Shevchenko University, Department of Physics, 2 Acad. Glushkova Pr., Kyiv,

03127, Ukraine for Scintillation Materials of STC “Institute for Single Crystals”, NAS of Ukraine, 60

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cInstitute

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Nauky Ave., 61001 Kharkiv, Ukraine

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E-mails:

[email protected] (Leonid Bulavin)

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[email protected] (Nickolai Lebovka); [email protected](L.N. Lisetski); [email protected] (L. Mazur);

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[email protected] (N. Vygornitskii)

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Received 1 August, 2018

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Contact information about Corresponding Author: Prof. L. N. Lisetski

Institute for Scintillation Materials of STC “Institute for Single Crystals”, NAS of Ukraine,

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60 Nauky Ave., 61001 Kharkiv, Ukraine

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E-mail address: [email protected](L.N.



Corresponding author.

E-mail address: [email protected](L.N. Lisetski)

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Lisetski);

ACCEPTED MANUSCRIPT ABSTRACT

Monte Carlo model for simulation of light transmission in anisotropic suspensions filled by absorbing cylindrical particles with anisotropic shape was developed. Aspect ratio of the cylinders (i.e., ratio of length and diameter) was varied between r=0 (infinitely thin disks) and r (infinitely thin rods). The cooperative filling model was used for simulation of clustering

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of particles controlled by aggregation parameter f. Orientation ordering of particles in anisotropic suspension was characterized by order parameters S. The effective cross section of

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particles and deviations from the Beer–Lambert–Bouguer law were studied as function of r, f and S. Obtained data may be useful for understanding the experimental data on optical

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properties of media filled with carbon nanotubes.

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transmission; Beer–Lambert–Bouguer law

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Keywords: Monte Carlo model, anisotropic suspensions, cylindrical particles, optical

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ACCEPTED MANUSCRIPT 1 Introduction Exponential Beer–Lambert–Bouguer law (hereafter the BLB law) describes the attenuation of light inside material owing to the presence of the absorbing species [1,2]

Tr  exp( zC)  exp( z /  ) .

(1)

Here Tr = I/I0, is the optical transmission, I0, I are the intensities of incident and transmitted light, respectively, z is the thickness of the material,  is the effective attenuation cross section

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of the absorbing particles, and C is a number concentration of particles (number of particles per unit volume). The value of  corresponds to the mean free path of the photons, and the

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value of =zC is called as the optical density.

This law is ubiquitous in nature and it is widely used for analysis of transmission of

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radiation in different fields including atmospheric, biological, colloidal and cosmological phenomena [3–5]. However, this law has many restrictions, especially at high concentrations

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of the absorbing species.

Several theoretical [6–9] and computational [10–14] studies of this problem have been

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performed. It has been demonstrated that the classical exponential extinction law follows from Poisson distribution of probabilities of absorption of photons for statistical independent events of absorption [7]. This exponential law also remains applicable to weakly inhomogeneous

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particulate media [8]. However, in highly heterogeneous media with correlated positions of the

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absorbing species and departure from spatial randomness the BLB law can be violated. The spatial correlations can result in formation of clusters of particles or creation of empty spaces and be manifested in form of the geometric shadowing effects [10]. In particular, it was

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demonstrated that effect of turbulent clustering in clouds and formation of void regions can cause increase in the direct transmittance [15].

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The detailed theoretical analysis of the effects of different types of spatial correlations between perfectly adsorbing spherical particles on the violations of the BLB law has been presented [6]. The media with negative, random (Poisson) and positive correlations were analysed. For negatively correlated media with more homogeneous distribution than for the Poisson ones the acceleration of the exponential attenuation (super- Poisson behaviour) was observed. For positively correlated media with cluster-like distribution the retardation of the exponential attenuation (sub-Poisson behaviour) was observed. Monte Carlo (MC) simulations, performed on the base of the “ballistic photons” model, confirmed the theoretical calculations in both regimes [11]. Negatively correlated media were simulated by random placing of particle at the distance not smaller than a specified distance away from any other 3

ACCEPTED MANUSCRIPT previously placed particle. Positive correlations were achieved by using a biased distribution to select a random distance from the previously placed particle to the next particle’s location. The more detailed MC simulations have been also performed for Poisson, clustered and fractal special distributions of particles [16]. The BLB law is widely used for characterization of suspensions of carbon nanotubes [17]. Significant violations of the BLB law has been observed in ordered liquid crystalline

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(LC) suspensions filled with nanoparticles (carbon nanotubes and inorganic platelets) [18,19]. The violations can reflect formation of ordered interfacial LC-layers near the surface of the nanoparticles and aggregation of nanoparticles in LC suspensions. The application of the BLB

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law to optically anisotropic systems was analyzed for the cases of randomly oriented particles,

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particles oriented in a stacks of planes and ideally oriented particles [20]. The significant violations of the BLB law were predicted for oriented groups of optically anisotropic particles.

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However, as far as we know, the detailed analysis of optical density of suspensions filled by oriented particles with anisotropic shape was never performed before. This work is devoted to the Monte-Carlo studies of optical transmission of suspensions filled by ideally

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absorbing cylinders. The cases isotropic and anisotropic orientations of particles as well as homogeneous and correlated spatial positions were analyzed. The rest of the paper is organized

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as follows. In Section 2, the computer simulation model and technical details of calculations

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are given. Section 3 presents main results and discussion. Section 4 presents conclusions. 2 Model and calculation details For convenience, in simulation procedure the supporting cubic lattice of size LLz

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with the unit-sized cells was used in our model. The particles randomly occupied the cells and the light beam propagates along z-axis. The periodical boundary conditions were used in all

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directions. The number concentration was defined as C=N/(L2z), where N is a total number of particles. The particles have the shape of cylinders with diameter of d and height of h. Aspect ratio was determined as r=h/d. The values of r0 or r corresponds to the infinitely thin disks or rods, respectively. The maximum size of the cylinder was put to be equal to the size of the lattice cell (i.e. d=1 at r1 (disk) and h=1 at r>1 (rod)). The cross section of the absorbing particle  depends upon its orientation with respect to the optical axis (polar angle). For example,

 =d2/4

(2a)

 =hd=h2/r

(2b)

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ACCEPTED MANUSCRIPT for orientation of particle along (=0) and perpendicular (=/2) to the optical axis, respectively. At r=1 the maximum value  =1 corresponds to the perpendicular orientation with respect to the optical axis. For disks (r1) the value of  varies in the interval from r (=/2) to /4 (=0). For rods (r>1) the value of  varies from /4r2(=0) to 1/r(=/2). The different anisotropic orientations of the cylinders were assumed. To characterize

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the uniaxial orientation the Herman’s orientation factor or orientation order parameter defined as [21]

(3)

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S==<3cos2-1>/2,

was used. Here P2() is the Legendre polynomial of degree 2 and <…> corresponds to the

and ideal orientation along optical axis, respectively.

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averaging over all particles. The values of S=0 and S=1 correspond to the random orientations

t

 (3 cos S

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1

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For realization of a desired orientation distribution of cylinders with order parameter of   1)d cos  ,

t

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2  d cos 

(4)

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the values of cos were uniformly distributed in the interval [1, t], where t  ( 8S  1  1) / 2 .

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For generation of correlated spatial distribution of cylinders an interactive cluster-

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growth model was used [22]. The special distribution is controlled by the value aggregation parameter f. The minimum f=0 corresponds to the stochastic distribution of particles, and increasing of f corresponds to the increasing of aggregation tendency (or positively correlated

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spatial distribution in the terminology used in [11]). The probability of occupation of cells was dependent on the number of occupied neighboring cells. The cell was occupied with probability

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1 in presence of occupied neighboring cells and with probability 10-f in absence of them. Figure 1 presents example of spatial distributions for randomly oriented disks and different values of f.

For calculation of an optical transmission Tr the effective cross section area unfilled by cylinders in each column of cells along z-axis was calculated. This procedure is schematically demonstrated for one column in Fig. 2 for disks (a) and rods (b). Finally, the calculated values of Tr were averaged over all LL columns.

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ACCEPTED MANUSCRIPT In calculation we use L=32, z=1-32 and the quantities under consideration were averaged over 100 independent statistical runs, unless otherwise explicitly specified in the text. 3 Results and their discussion The optical transmission, Tr, can be easily estimated for a model in which the particle occupies the centers of the cubic lattice and totally cover the unit-sized cells. In that case the probability of finding of an empty column with thickness of z is (1-С)z and

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Tr =(1-C)z. For particles with cross section area of  we have

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Tr =(1-C)z.

(5a)

(5b)

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In the limit of very small concentration С<<1, Eq. (1) and Eq. (5b) are equivalent. The parameter =1-Tr1/z=C (hereafter the violation parameter) does not depend upon

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the thickness of the absorption layer, z, and it can useful for checking of the violation of the BLB law.

Figure 3 presents examples of the optical transmission Tr(a) and violation parameter (b)

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versus the thickness of absorption layer, z, dependences at different values of aggregation parameter, f. These data were obtained by simulations at fixed number concentration, С=1,

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order parameter, S=0, and aspect ratio, r=5. Near linear lnTr(z) dependences were only observed for the stochastic (f=0) and weakly aggregated (f=3) distributions (Fig. 3a). It

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indicated in favour of the classical exponential extinction BLB law. However, in highly aggregated suspensions at f=5 the significant deviation from linear lnTr(z) dependence was observed. These data were in qualitative correspondence with previously reported results for

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spherical species [7,8]. Note that (z) dependences also revealed violations of BLB law for relatively weakly aggregated suspensions (Fig. 3b). For example, for f=3 the value of violation

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parameter, , initially increased with growth of z and then saturated for thick layers (z>5-10). For highly aggregated suspensions at f=5 the (z) dependences were even more complicated. The observed violations of BLB law for aggregated suspensions evidently reflect the geometric shadowing effects [10]. The effects can reflect the structure of clusters and empty spaces. In present work the simplified interactive cluster-growth model was used for simulation of aggregation of cylinders [22]. For this model the observed anomalies at small values of Z (z<10) for f=3 and f=5 (Fig. 3b) can also reflect structural features of clusters formed at the applied periodic boundary conditions. In general case we can speculate that the violations of

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ACCEPTED MANUSCRIPT BLB law can be governed by the type of particle aggregation model [23,24] and boundary conditions. The obtained data evidenced that violation of BLB law is a more typical rule for aggregated suspensions. For definiteness hereafter the data obtained for relatively thick absorption layer (z=32) are only presented. Figure 4 presents the optical transmission Tr versus the number concentration of

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particles, C, for non-aggregated (f=0, a) and strongly aggregated (f=5, b) suspensions. The data were presented for random orientation of particles (S=0). For non-aggregated suspensions (f=0, a) the observed linear lnTr(z) dependences indicated in favour of the classical exponential

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extinction BLB law. The violations of the BLB law were observed strongly aggregated

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suspensions (f=5, b). The most significant violations were observed at r=1, but they became less prominent for large values of r, i.e. for elongated rods.

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However, the initial parts of lnTr(z) dependences at very small concentrations (C0) were always linear (Fig. 4) and they were used to estimate the effective cross section area of particles, .

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Figure 5 presents examples of cross section relative value, *, versus particle aspect ratio, r, for non-aggregated (f=0, a) and strongly aggregated (f=5, b) suspensions. Here,

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*=/ where   is the theoretical cross section for particle with orientation along (=0) the

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optical axis (i.e., at S=1, Eq. (2a)). At r<1 (i.e., for disks) the dependencies *(r) were approximately linear for the non-aggregated (f=0, a) suspensions and non-linear for the strongly aggregated suspensions (f=5, b). At r1 (i.e., for cylinders), for both the non-aggregated and

approximated as

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strongly aggregated suspensions the dependencies *(r) were nearly linear and can be well

* = a(r-1)+ *1 at r1,

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(6)

where a (slope) and *1=*(r=1) are the fitting parameters. The coefficients of determination of the fitting were rather high, R2>0.97. Figure 6 presents the dependences of parameters *1 (a) and a (b) versus order parameter, S. The slopes of the *(r) curves were maximums for the non-aggregated suspensions, they decreased with increasing of S and tend to the zero for perfectly oriented suspensions (at S=1) (Fig. 6a). The value of *1 was nearly independent of order parameter for weakly oriented suspensions (S<0.5), but it decreased with increasing of S for more perfectly oriented suspensions. In addition, *1 decreased with increase of aggregation degree, f.

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ACCEPTED MANUSCRIPT For practically interesting case of the very long rods (e.g., for carbon nanotubes) when r>>1 and completely disordered suspensions (S=0) we have for the effective cross section of particles

  a(f)dh /4,

(7)

where value of a(f)(1.064-0.010f2) decreases with increasing of aggregation degree. So, the cross section  is proportional to the product of particle dimensions d.h and

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impact of the aggregation degree, f, evidently reflects the effects of particle shading. Moreover, the value of  could significantly decrease in presence of orientation ordering, for example, for

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rods dispersed in liquid crystalline suspensions.

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4 Conclusions

The Monte Carlo analysis of optical density behavior of suspensions filled by oriented

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particles with anisotropic shape was performed. The cases of isotropic and anisotropic orientations of particles as well as homogeneous and correlated spatial positions were analyzed. The constancy of violation parameter =1-Tr1/z at different thickness of the absorption layer

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was tested for different aggregated systems. For non-aggregated suspensions the classical exponential extinction BLB law was observed. However, the highly aggregated suspensions

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demonstrated rather complicated (z) dependences and obtained data evidenced that the

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violation of BLB law is a more typical rule for aggregated suspensions. So, for comparison of data for such suspensions it is necessary to fix the thickness of the absorption layer. The level of violation of BLB law was dependent upon particle aspect ratio and it became less prominent for elongated rods, i.e. at large aspect ratio, r>>1. For elongated rod the effective cross section

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area of particles, , grown linearly with increasing aspect ratio and impact of the aggregation degree, f, on the value of  evidently reflected the effects of particle shading. Moreover, the

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value of  could significantly decrease in presence of orientation ordering (for rods dispersed in liquid crystalline suspensions). The predicted complex effects of clustering and orientation ordering can be useful for explanation of recently reported behaviour of optical density of liquid crystalline suspensions filled with carbon nanotubes [25–30]. Acknowledgements This work was partially funded by a joint project of Department of Targeted Training of Taras Shevchenko National University of Kyiv at the National Academy of Sciences of Ukraine, project # 15F (0117U006352).

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ACCEPTED MANUSCRIPT Figure captions Fig. 1. Example of spatial distributions for randomly oriented disks (r=0.2, S=0) and different values of f. The lattice size was 323232, and number concentration was С=0.1.

Fig. 2. To description of procedure of calculation of the effective cross section area unfilled by

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cylinders. The one column of cells along z-axis is shown for r=0.2 (disks, a) and r=2 (rods, b). Fig. 3. Optical transmission, Tr, (a) and violation parameter, , (b) versus the thickness of Here, the number

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absorption layer, z, at different values of aggregation parameter, f.

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concentration is С=1, order parameter is S=0, and aspect ratio is r=5.

Fig. 4. Optical transmission Tr versus the number concentration of particles, C, for non-

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particles, r. Here, the order parameter is S=0.

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aggregated (f=0, a) and strongly aggregated (f=5, b) suspensions and different aspect ratio of

Fig. 5. Cross section relative value, *, versus particle aspect ratio, r, for non-aggregated (f=0, a) and strongly aggregated (f=5, b) suspensions. Here, *=/  where   is the theoretical

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cross section for particle with orientation along the optical axis (Eq. (2a). The data different

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order parameters, S=0-1 are presented. Insets show initial parts of *(r) at r1. Fig. 6. Parameters *1 (a) and a (b) (Eq. 6) versus order parameter, S, for different values of

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parameter of aggregation, f.

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ACCEPTED MANUSCRIPT Highlights  Monte Carlo simulation of light transmission in anisotropic suspensions was performed  Cylindrical particles with different aspect ratio (disks and rods) were considered  Clustering and orientation ordering of particles were taken into account

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 Cross section of particles and deviations from the Beer–Lambert–Bouguer law were evaluated

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Figure 1

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Figure 6