Refractive index and polarizability of polystyrene

J Mater Sci POLYMERS Polymers

Refractive index and polarizability of polystyrene under shock compression Xuping Zhang1 , Guiji Wang1,* , Binqiang Luo1 Chengwei Sun1 , and Cangli Liu3

, Fuli Tan1

, Simon N. Bland2

, Jianheng Zhao1,*

,

1

Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621999, China Blackett Laboratory, Imperial College London, London SW72AZ, UK 3 China Academy of Engineering Physics, Mianyang 621999, China 2

Received: 12 March 2018

ABSTRACT

Accepted: 22 May 2018

The refractive index, polarizability and thermodynamic response of polystyrene under shock compression were investigated through experiments and theoretical analysis, and a relationship between the refractive index and the density, pressure and temperature of the polystyrene was obtained. Above a pressure of 20 GPa, an obvious inflexion was observed in how the refractive index of the polystyrene varied with the thermodynamic variables; in particular, it was found to depend strongly on the temperature (as obtained using a semiempirical complete equation of state). Relating the measured refractive index to the polarizability indicates that the polarizability decreased from 1.28 9 10-23 cm3 at ambient conditions, to 0.98 9 10-23 cm3 at pressure of 33 GPa, indicating that the lowest direct band gap Et of polystyrene becomes \ 2 eV, similar to that of many semiconductor materials.

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Media, LLC, part of Springer Nature 2018

Introduction Insight into the electrical properties and the equation of state (EOS) of hydrocarbons under high pressure is significant for studying the planetary carbon, formation of organic materials in the early solar system and formation of planetary magnetic fields [1–4]. Polystyrene (PS) is often used as the sample in highpressure studies, due to it being composed of purely of hydrogen and carbon. Additionally, PS is also used as an ablator material for laser-based high energy density physics experiments due to its easy to make and shape accurately and because researchers have

the ability to controllably introduce dopant atoms that are chemically bonded to hydrogen [5–7] forming tracer layers for spectrographic diagnostics. Understanding the thermodynamical and electrical properties of PS is therefore critical for validating the models in a wide variety of fields [5–7]. Measurements of the reflectivity/refractivity of materials are an effective and popular method for obtaining the electrical properties of materials under high pressure (including measurements of metallization) [8–10]. The reflectivity and electrical conductivity of PS under high pressure were studied experimentally by M. Koenig et al. in 2003 [11], Ozaki

Address correspondence to E-mail: [email protected]; [email protected]

https://doi.org/10.1007/s10853-018-2489-8

J Mater Sci

et al. in 2009 [12] and M. A. Barrios et al. in 2010 [13]. Due to the differences in diagnostic configurations or from probe beam stability, the results of these experimental studies appear to be somewhat divergent. M. Koenig reported that PS undergone a drastic increase in reflectivity at around 80 GPa and saturate at 50% around 170 GPa [11]. Ozaki et al. [12] argued that reflectivity of PS from 16 to 42% in pressure ranges 300–500 GPa. And M. A. Barrios et al. [13] indicated that PS become reflective when shocked to 100–200 GPa and have a reflectivity of 40% above 300 GPa. Recently, the reflectivity of PS was also studied through molecular dynamic simulations [14]. Though the dissociation and metallic process were systematically discussed, the pressure range and optical properties related to dissociation were not still coincident with experiments [11–14]. All above works focused on optical properties or metallic transition in pressure range of several hundreds of GPa. The shock Hugoniot of PS, however, shows that there is a clear volume change and inflexion at pressure of * 20 GPa [15–17]. Both statistical mechanical models and molecule simulations [18–21] require detailed knowledge of the properties of PS at this inflexion point for them to accurately predict dissociation and metallic transition at higher pressures; but up to now few works have examined the refractive index, and hence electrical properties, at * 20 GPa. Further, at shock pressures of * 20 GPa the temperatures formed in experiments are \ 2000 K, which is very difficult to measure experimentally in a [22, 23], limiting any studies of the effect of temperature on the refractive index here. The objectives of this paper are to obtain the refractive index, polarizability and thermodynamical response of polystyrene. The findings in this work will be a basis for validating the materials models and molecular dynamic simulations, enabling better designs for astrophysics and laboratory-based HEDP applications. The structure of the rest of this paper is as follows. Experimental techniques and conditions are described in ‘‘Experimental techniques and conditions’’ section. The refractive index and thermodynamical responses of PS are calculated and discussed in ‘‘Refractive index of PS’’ section, followed by the polarizability of PS in ‘‘Polarizability of PS’’ section. Finally, we close our paper with concluding remarks in ‘‘Conclusion’’ section.

Experimental techniques and conditions Shock compression was used to generate high pressures, densities and temperatures in the PS samples. The schematic diagram of experimental configuration is shown in Fig. 1a. Shock waves were generated through flyer plate impact, with the plates launched by compact pulsed power generator, CQ4. A multiMA current produced by CQ4 flowed through a pair of electrode panels short-circuited on one end, and interacted with the self-induction magnetic fields between the electrode panels, launching a section of the panels to very high velocity, while maintaining a solid leading surface with extremely good flatness (the detailed principle and optimizations of magnetically driven flyer plates on CQ4 can be found in Ref. [24–26]). The PS sample was backed by LiF window to enable both shock and re-shock data to be obtained in 1 experiment, allowing the thermodynamical parameters of PS to be obtained more readily. A Dual Laser Heterodyne Velocimetry (DLHV) system, with accuracy of 1% [27], monitored the velocity of both the flyer–PS and the PS–LiF interfaces. The wavelength of DLHV is 1550 nm. The flyer plate was flat to * 1 lm with a 20-nm surface finish and was parallel to within * 3 lm. The initial density of the PS was 1.05 g cm-3. PS samples for these experiments were diamond-turned to final dimensions of 6–10 mm diameter with 0.26–1 mm thickness. Typical velocity and frequency spectrums from an experiment are shown in Fig. 1b and c. The analyses of shock wave propagation and interaction are exhibited in Fig. 1d for helping to understand velocity profiles. The flyer velocity, shock wave transition time and apparent particle velocity in PS were directly obtained from the velocity profiles. The detailed experimental conditions of flyer plate and samples are summarized in Table 1.

Refractive index of PS The refractive index under shock loading [28] was determined using n¼

n0 D  ua ; Du

ð1Þ

where n0 is the refractive index of PS at ambient conditions, D is shock wave velocity, u is particle

J Mater Sci

(a)

(b)

velocity/(km s-1)

3

CQ4-shot-584

flyer veocity 2

1

Flyer/PS interface (apparent velocity) PS/LiF interface (apparent velocity)

0 3.6

3.8

4.0

4.2

4.4

4.6

4.8

t/(μs)

(c)

(d)

Figure 1 a Schematic diagram of experimental setup. b Typical velocity profiles in experiments (here the velocity of CQ4-shot-584 is taken as an example). c Frequency spectrum of CQ4-shot-584. d X-t paragraph (not to scale), X-axis is the Lagrangian

coordinate. ‘v1’ denotes the velocity of flyer/PS interface, ‘v2’ denotes the velocity of PS/LiF interface, ‘0’–‘5’ denotes the time point correlated with velocity profiles.

velocity, and ua is apparent particle velocity. Here, n0 was determined by experimental results and Cauchy formula in ‘‘Determine the refractive index of PS at ambient conditions’’ section . The shock wave velocity D and apparent particle velocity ua were obtained from the velocimetry data. The particle velocity u was determined from the shock Hugoniot of flyer plate, flyer plate velocity and initial density of the sample by means of impedance matching method. Measurements of both the shock and re-shock(s) were used to determine the thermophysical parameters of the PS. Using these, a reliable complete EOS was established for obtaining the shock temperature for estimating the temperature effect on refractivity (see ‘‘Complete EOS and shock temperature’’ section).

Determine the refractive index of PS at ambient conditions The refractive index at different wavelengths at ambient conditions was measured by Abbe refractometer with instrument ATAGO DR-M2. The data of refractive index are fitted by Cauchy formula [29] n¼aþ

b c þ 4; 2 k k

ð2Þ

where a, b and c are constant, n is refractive index, and k is wavelength. The experimental results and the fitting line of refractive index of PS at ambient conditions are demonstrated in Fig. 2. When a = 1.56, b = 1.3 9 104 (nm2), c = - 5 9 108 (nm4), fitting line of Cauchy formula describes the experimental results [13, 30, 31] well. The determined refractive index n0 at 1550 nm was 1.565.

J Mater Sci

Table 1 Experimental conditions Shot no.

Flyer plate material

Flyer plate sizes (mm)

Sample sizes (mm)

Flyer velocity (km/s)

DPS (km/s)

ua (km/s)

Ua2 (km/s)

CQ4-shot-495

Al

26 9 12 9 1.000

CQ4-shot-656 CQ4-shot-686 CQ4-shot-688 CQ4-shot-584

Al Al Al Cu

26 12 12 26

CQ4-shot-653

Cu

26 9 12 9 1.001

CQ4-shot-658 CQ4-shot-659

Cu Cu

12 9 10 9 0.800 20 9 8 9 0.800

S1 U10 9 0.556 S2 U10 9 0.855 12 9 8 9 0.558 U8 9 0.260 U8 9 0.261 S1 U8 9 0.546 S2 U8 9 0.546 S1 U8 9 1.068 S2 U8 9 1.066 U10 9 0.510 U8 9 0.511

4.203 4.152 5.610 4.354 5.650 1.713 1.739 1.059 1.036 2.429 2.579

6.78 6.884 7.75 6.842 7.5 4.986 4.946 3.954 3.807 6.069 6.197

2.969 2.947 4.304 3.164 4.394 1.444 1.464 0.913 0.894 2.043 2.131

2.439 2.340 3.536 2.500 3.583 1.086 1.104 – – 1.679 1.805

9 9 9 9

8 9 1.000 8 9 0.800 8 9 0.800 12 9 1.002

Flyer velocity denotes the velocity when the flyer plate impacts the PS sample. Because the flyer plate is accelerated gradually to high velocity in a short flight distance (the initial distance between flyer and sample) for magnetically driven flyer plates, sometimes the velocity of flyer plates is sensitive to flight distance of flyer [25]. Hence, the difference of flyer plate impact velocity in one experiment is caused by the flyer distance and flyer plate planarity. S1 denotes sample 1. S2 denotes sample 2. ‘–’ denotes empty. ‘ua’ is apparent particle velocity of flyer/PS interface. ‘DPS’ is shock velocity of the first wave. ‘ua2’ is apparent particle velocity of PS/LiF interface

1.66 Ref. [30] Ref. [13,31]

Refractive index

1.64

fit the empirical Cauchy formula this work

1.62

Refractive index at 1550nm

1.60

1.58

1.56 200

400

600

800

1000 1200 1400 1600 1800

wavelength/(nm) Figure 2 Refractive index of PS at ambient condition.

The shock Hugoniot and re-shock data The shock wave velocity D was determined by the thickness of samples divided by the transition time in sample. The particle velocity u was determined by using impedance matching method with the shock Hugoniot of flyer plate, flyer plate velocity and initial density of samples. Here, the densities of copper and aluminum flyer plates were 8.93 and 2.712 g cm-3, respectively. The shock Hugoniot of copper D = 3.94 ? 1.489u and aluminum D = 5.38 ? 1.34u, where D is the shock wave velocity and u is the particle velocity [32], was used in our calculations.

The apparent velocity ua was measured directly by DLHV. The re-shock states were determined from the shock Hugoniot of LiF and the PS/LiF interface velocity by using impedance matching method. The shock Hugoniot of LiF D = 5.148 ? 1.353u, where D is shock wave velocity and u is particle velocity [32–34], was used in our calculations. The density of LiF was 2.64 g cm-3. The refractive index and velocity correction of LiF were obtained from the references [33, 34]. The constant value of 1 ? (ua2- u2)/u2 = 1.274 was used for obtaining the particle velocity of PS/LiF interface velocity u2 according to the apparent velocity of PS/LiF interface velocity ua2. The pressure versus particle velocity of shock Hugoniot and re-shock data is shown in Fig. 3. The results display a linear D–u relation of PS given by D = 2.172 ? 1.7u for P \ 20 GPa and D = 2.7 ? 1.3u for P [ 20 GPa, respectively (Fig. 4a). The uncertainty of shock Hugoniot of PS is 1% for particle velocity, 1.5% for shock wave velocity and 2% for pressure by means of uncertainty transfer equation.

Complete EOS and shock temperature A semiempirical three-term complete EOS was constructed to obtain the thermodynamic behavior of polystyrene at high pressure and high temperature. According to lattice dynamics and electronic

J Mater Sci

excitations theory of solid physics [35, 36], the Helmholtz free energy is written as

120

shock reshock PS Hugoniot (P20GPa) D=2.7+1.3×u LiF Hugoniot D=5.148+1.353×u

100

P/(GPa)

80 60

FðV; TÞ ¼ FC ðV Þ þ Fa ðV; TÞ þ Fe ðV; TÞ;

40 20 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

u/(km/s) Figure 3 Shock Hugoniot and re-shock data of PS.

D shock wave velocity/(km/s)

(a) 30

G. E. Hauver LASL Shock data M. A. Barrios C. Wang A.V. Bushman this work, experiments this work, P20GPa

25 20 15 10 5

12 10 8 6 4 2 0

0 0

5

0

1

2

10

3

4

5

15

6

20

u particle velocity/(km/s)

(b) 500 G. E. Hauver LASL Shock data bushman this work,experiments M. A. Barrios C. Wang this work, P20GPa

Pressure/(GPa)

400 100

300

80 60

200

40 20

100 0 0.2

0

0.3

0.4

0.4 0.5 0.6 0.7 0.8

0.5

0.6

0.7

0.8

0.9

1.0

V/V0 Figure 4 Shock Hugoniot, a D–u relationship, b pressure–volume relationship.

ð3Þ

where FC ðV Þ is the cold energy which describes the potential (cold) part of the interaction at T = 0 K, Fa ðV; TÞ is the thermal contribution by atom, Fe ðV; TÞ is thermal contribution by thermal excitation of electrons, V is volume, and T is temperature. Other thermodynamic functions are obtained from free energy in the usual way. Specifically, the pressure is  oF  oðF=T Þ P ¼  oV and internal energy is E ¼  T2 oT V . T The Mie-potential was used in the calculation of cold part [37]. According to Debye model, thermal contributions of atoms were defined by the excitation of acoustic and optical modes of thermal vibrations of atoms [37]. The thermal contribution of electron was approximately calculated by free electron model [38, 39]. The details of EOS model and parameters determination can be seen in ‘Appendix I: Complete EOS OF PS.’ The complete EOS model and its parameters were validated by comparing the calculated shock Hugoniot with previous experimental results. The calculated shock Hugoniot curve is shown in Fig. 4, which agrees well with the experimental data from Ref. [12, 13, 15, 17] in pressure range of 500 GPa. According to the complete EOS model and Rankine– Hugoniot relations, we can obtain the shock temperature of PS. The dissociation temperature and pressure of PS were approximately calculated with free energy model by using a fix C–H bond dissociation energy 4.2 MJ/kg [40]. The calculated shock temperatures agree well with previous experiments and calculations (Fig. 5) [12–14, 41]. Based on the comparisons in Figs. 4, 5 and 11 (in ‘‘Appendix’’), the obtained complete EOS model and its parameters were proofed to be reliable. Hence, the shock temperature which calculated from EOS can be used for estimating the temperature effect on refractivity in the following. According to the experimental conditions in Table 1 and related calculation formulas, the physical results of shock Hugoniot, re-shock data, refractive index and shock temperature are derived and summarized in Table 2.

J Mater Sci

this work P20GPa C. Wang M. A. Barrios dissociation F. H. Ree SESAME 7590 SESAME 7592 N. Ozaki S. X. Hu

Temperature/(kK)

40

30

20

8

dissociation

6 4

10

2 0 0

0 0

100

200

20 40 60 80 100

300

400

500

Pressure/(GPa)

density at pressure above 20 GPa. The relationship of refractive index versus pressure is also shown in Fig. 6b. The change trend of refractive index with pressure and density is similar. The relationship of refractive index versus temperature is shown in Fig. 7. An obvious nonlinear inflexion is observed at temperature around 2100 K. At this pressure and temperature, PS starts dissociation. A relative slope change (RSC) of refractive index was defined for analyzing the dependence of refractive index on density, pressure and temperature RSCðxÞ ¼

Figure 5 Shock temperature of PS.

Relationships of refractive index with density, pressure and temperature The refractive index at high pressure and its uncertainty is shown in Fig. 6. The uncertainty transfer equation of refractive index is deduced from Eq. (1) and shown in ‘Appendix II: Uncertainty analysis.’ The maximum uncertainty of refractive index is 1.6% in experiments. The relationship of the refractive index n with compression ratio below 20 GPa is linear, with n given by n ¼ 0:9838 þ 0:587q=q0 ;

ð4Þ

where q is density and q0 is initial density. The slope of reflective index is changed at compression ratio of about 1.8 and pressure of around 20 GPa. The increase in refractive index becomes slowly with the

ðdn=dxÞ2  ðdn=dxÞ1 ðdn=dxÞ1

ð5Þ

where x is free variation for density, pressure and temperature. The RSC at pressure above 20 GPa is approximately - 0.528 for density, - 0.524 for pressure and larger than 10.7 for temperature. The trend of refractive index dependent on pressure and density is similar. The refractive index at pressure above 20 GPa increases slower than that at pressure below 20 GPa. Though it is crude, it is obvious that the dependence on temperature increases rapidly at pressure above 20 GPa.

Polarizability of PS The polarizability of PS was determined by Lorentz– Lorenz equation [42]

Table 2 Summary of the deduced physical results Shot no.

u (km/s)

Shock pressure (GPa)

q/q0

Shock temperature/(kK)

Refractive index

u2 (km/s)

Re-shock pressure (GPa)

CQ4-shot-495

3.063 2.998 4.01 3.157 4.072 1.505 1.530 0.951 0.933 2.093 2.218

21.8 21.7 32.6 22.7 32.1 7.88 7.94 3.9 3.7 13.3 14.4

1.824 1.772 2.072 1.857 2.188 1.432 1.448 1.317 1.325 1.526 1.557

2.115 2.093 2.165 2.184 2.136 1.019 1.031 0.686 0.668 1.464 1.564

2.056 2.014 2.092 2.047 2.142 1.827 1.837 1.757 1.762 1.875 1.902

1.913 1.835 2.773 1.968 2.81 0.853 0.866 – – 1.317 1.416

39.1 37.0 65.2 40.5 66.4 14.2 14.4 – – 24.1 26.4

CQ4-shot-656 CQ4-shot-686 CQ4-shot-688 CQ4-shot-584 CQ4-shot-653 CQ4-shot-658 CQ4-shot-659

‘u’ is the particle velocity of PS. ‘u2’ is the particle velocity of PS/LiF interface. ‘–’ denotes empty

J Mater Sci

(a)

(b) 2.4

2.4

Refractive index n=0.9838+0.587 ×ρ/ρ0 n=1.528+0.277 × ρ/ρ0

2.2

Refractive index

Refractive index

Figure 6 Refractive index of PS at high pressure. a n–q, b n–P.

2.0 1.8

1.0

1.2

1.4

1.6

1.8

ρ/ρ0

2.4 Refractive index n= 1.632+0.187 × T n= -2.6+2.2 × T

Refractive index

2.0

1.8

1.6 0.5

1.0

1.5

2.0

2.5

Temperature/(kK) Figure 7 Refractive index–temperature relationship.

  ðn2  1Þ 4p NA Pa ¼ ; ðn2 þ 2Þq 3 l

ð6Þ

where NA is Avogadro’s number. Pa is polarizability. lis the molecular weight. Refractive index below the interband absorption edge is analyzed by using a single effective oscillator fit [43] n2  1 ¼

Ed E0 E20  h2 x2

ð7Þ

where hx is photon energy, E0 is the single oscillator energy, and Ed is the dispersion energy. The ambient values of Ed and E0 were determined by fitting the refractive index to measure values in the range from 800 to 1800 nm, where the wavelength region near the probe laser. The obtained Ed is 9.7 eV and E0 is 6.8 eV. The lowest direct band gap Et is related empirically to the oscillator energy E0 by [43] Et ðeVÞ ¼ E0 =1:5:

2.2 2.0 1.8 1.6

1.6

2.2

Refractive index n=1.70561+0.01475 × P n=1.88956+0.00702 × P

ð8Þ

2.0

2.2

2.4

0

5

10

15

20

25

30

35

40

P/(GPa)

The relationships of polarizability and the lowest direct band gap with density are shown in Fig. 8. The uncertainty transfer equation of polarizability and the lowest direct band gap is deduced from Eqs. (6) and (7), respectively, and shown in ‘Appendix II: Uncertainty analysis.’ The maximum uncertainty of polarizability is 3.4% in experiments. Because the uncertainty of dispersion energy uses 8% according to the difference between experimental value and empirical estimated value of organic liquid [43], the deduced maximum uncertainty of the lowest direct band gap is 9.2%. The polarizability and lowest direct band gap in PS gradually decrease (causing the refractive index increase) as the molecular distance reduces due to compression. The slopes of the polarizability and the lowest direct band gap both change at pressure above 20 GPa. If the change of lowest direct band gap at 20 GPa is ignored, both the extrapolation of band gap in Fig. 8 and the Goldhammer–Herzfeld criterion [44, 45] indicate that the band gap of PS will still not have closed at pressures above * 1TPa, i.e., the PS will not be fully metalized. In reality, the dissociation of PS starts at pressure 20 GPa and the metallization of PS at higher pressures should be reconsidered. The polarizability decreases from 1.28 9 10-23 cm3 at ambient condition to 0.98 9 10-23 cm3 at pressure up to 33 GPa. The value of lowest direct band gap Et of PS at pressure above 20 GPa is less than 2 eV, which the values approximately approach to most semiconductors’ [39]. The relationships of polarizability and the lowest direct band gap with temperature are shown in Fig. 9. The polarizability and lowest direct band gap decrease rapidly with temperature around 2100 K. It can be concluded that the polarizability and lowest direct band gap are more sensitive to temperature by

5 Polarizability Et lowest direct bandgap guiding line (P