Abstract

Terahertz (THz) radiation interacts strongly with the intermolecular hydrogen-bond network in aqueous liquids. The dielectric properties of liquid water and aqueous solutions in the THz spectral region are closely linked to the microscopic dynamics of the liquid solution, and hence THz spectroscopy offers an important insight into fundamental intermolecular interactions in polar liquids. At the same time, the strong and characteristic interaction between THz radiation and liquids offers a methodology for the classification of liquids inside containers, and hence the THz region is suitable for remote detection of some of the properties of bottled liquids. Here we present a review of THz spectroscopy and modeling of water–ethanol mixtures, and establish a link between the dielectric function of water–ethanol mixtures and some of their thermodynamic properties. We then review how the knowledge of the dielectric function of aqueous mixtures can be used for inspection of liquids inside bottles. Finally we draw up some of the limits to the applicability of THz reflection spectroscopy in the identification of dangerous liquids.

© 2009 Optical Society of America

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2009

H. Yada, M. Nagai, and K. Tanaka, “The intermolecular stretching vibration mode in water isotopes investigated with broadband terahertz time-domain spectroscopy,” Chem. Phys. Lett. 473, 279-283 (2009).
[CrossRef]

2008

T. Arikawa, M. Nagai, and K. Tanaka, “Characterizing hydration state in solution using terahertz time-domain attenuated total reflection spectroscopy,” Chem. Phys. Lett. 457, 12-17 (2008).
[CrossRef]

M. Heyden, E. Bründermann, U. Heugen, G. Niehues, D. M. Leitner, and M. Havenith, “Long-range influence of carbohydrates on the solvation dynamics of water--Answers from terahertz absorption measurements and molecular modeling simulations,” J. Am. Chem. Soc. 130, 5773-5779 (2008).
[CrossRef] [PubMed]

H. Yada, M. Nagai, and K. Tanaka, “Origin of the fast relaxation component of water and heavy water revealed by terahertz time-domain attenuated total reflection spectroscopy,” Chem. Phys. Lett. 464, 166-170 (2008).
[CrossRef]

P. Uhd Jepsen, J. K. Nielsen, and U. Møller, “Characterization of aqueous alcohol solutions in bottles with THz reflection spectroscopy,” Opt. Express 16, 9318-9331 (2008).
[CrossRef] [PubMed]

2007

P. Uhd Jepsen, U. Møller, and H. Merbold, “Investigation of aqueous alcohol and sugar solutions with reflection terahertz time-domain spectroscopy,” Opt. Express 15, 717-737 (2007).
[CrossRef]

S. Schrödle, B. Fischer, H. Helm, and R. Buchner, “Picosecond dynamics and microheterogenity of water+dioxane mixtures,” J. Phys. Chem. A 111, 2043-2046 (2007).
[CrossRef] [PubMed]

2006

M. Nagai, H. Yada, T. Arikawa, and K. Tanaka, “Terahertz time-domain attenuated total reflection spectroscopy in water and biological solution,” Int. J. Infrared Millim. Waves 27, 505-515 (2006).
[CrossRef]

U. Heugen, G. Schwaab, E. Bründermann, M. Heyden, X. Yu, D. M. Leitner, and M. Havenith, “Solute-induced retardation of water dynamics probed directly by terahertz spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 12301-12306 (2006).
[CrossRef] [PubMed]

2005

H. Kitahara, T. Yagi, K. Mano, M. W. Takeda, S. Kojima, and S. Nishizawa, “Dielectric characteristics of water solutions of ethanol in the terahertz region,” J. Korean Phys. Soc. 46, 82-85 (2005).

C. Zhang and X. Yang, “Molecular dynamics simulation of ethanol/water mixtures for structure and diffusion properties,” Fluid Phase Equilib. 231, 1-10 (2005).
[CrossRef]

A. A. Potapov and I. Yu. Parkhomenko, “Dielectric properties of solutions isomorphous with water,” Russian J. Gen. Chem. 75, 34-39 (2005).
[CrossRef]

P. Uhd Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30, 29-31 (2005).
[CrossRef] [PubMed]

H. Hirori, M. Nagai, and K. Tanaka, “Destructive interference effect on surface plasmon resonance in terahertz attenuated total reflection,” Opt. Express 13, 10801-10814 (2005).
[CrossRef] [PubMed]

2004

2002

J. E. Boyd, A. Briskman, C. M. Sayes, D. Mittleman, and V. Colvin, “Terahertz vibrational modes of inverse micelles,” J. Phys. Chem. B 106, 6346-6353 (2002).
[CrossRef]

2001

J. E. Boyd, A. Briskman, V. L. Colvin, and D. M. Mittleman, “Direct observation of terahertz surface modes in nanometer-sized liquid water pools,” Phys. Rev. Lett. 87, 147401 (2001).
[CrossRef] [PubMed]

D. C. Sorescu, B. M. Rice, and D. L. Thompson, “Molecular dynamics simulations of liquid nitromethane,” J. Phys. Chem. A 105, 9336-9346 (2001).
[CrossRef]

2000

P. Petong, R. Pottel, and U. Kaatze, “Water-ethanol mixtures at different compositions and temperatures. A dieletric relaxation study,” J. Phys. Chem. A 104, 7420-7428 (2000).
[CrossRef]

1999

T. Sato, A. Chiba, and R. Nozaki, “Dynamical aspects of mixing schemes in ethanol-water mixtures in terms of the excess partial molar activation free energy, enthalpy, and entropy of the dielectric relaxation process,” J. Chem. Phys. 110, 2508-2521 (1999).
[CrossRef]

H. E. Alper, F. Abu-Awwad, and P. Politzer, “Molecular dynamics simulations of liquid nitromethane,” J. Phys. Chem. B 103, 9738-9742 (1999).
[CrossRef]

1997

C. Rønne, L. Thrane, P.-O. Astrand, A. Wallqvist, K. V. Mikkelsen, and S. R. Keiding, “Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation,” J. Chem. Phys. 107, 5319-5331 (1997).
[CrossRef]

D. M. Mittleman, M. C. Nuss, and V. L. Colvin, “Terahertz spectroscopy of water in inverse micelles,” Chem. Phys. Lett. 275, 332-338 (1997).
[CrossRef]

1996

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321-2323 (1996).
[CrossRef]

J. T. Kindt and C. A. Schmuttenmaer, “Far-Infrared dielectric properties of polar liquids probed by femtosecond terahertz pulse spectroscopy,” J. Phys. Chem. 100, 373-379 (1996).
[CrossRef]

P. Uhd Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B 13, 2424-2436 (1996).
[CrossRef]

1995

L. Thrane, R. H. Jacobsen, P. Uhd Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330-333 (1995).
[CrossRef]

1991

J. M. Alison and R. J. Sheppard, “A precision waveguide system for the measurement of complex permittivity of lossy liquids and solid tissues in the frequency range 29 GHzto90 GHz--III. The liquid system for 57to82GHz: an investigation into water and formamide,” Meas. Sci. Technol. 2, 975-979 (1991).
[CrossRef]

1990

J. Barthel, K. Bachhuber, R. Buchner, and H. Hetzenauer, “Dielectric spectra of some common solvents in the microwave region. Water and lower alcohols,” Chem. Phys. Lett. 165, 369-373 (1990).
[CrossRef]

T. J. Parker, “Dispersive Fourier transform spectroscopy,” Contemp. Phys. 31, 335-353 (1990).
[CrossRef]

D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006-2015 (1990).
[CrossRef]

1989

M. van Exter, Ch. Fattinger, and D. Grischkowsky, “Terahertz time-domain spectroscopy of water vapor,” Opt. Lett. 14, 1128-1130 (1989).
[CrossRef]

U. Kaatze, “Complex permittivity of water as a function of frequency and temperature,” J. Chem. Eng. Data 34, 371-374 (1989).
[CrossRef]

S. Mashimo, S. Kuwabara, S. Yagihara, and K. Higasi, “The dielectric relaxation of mixtures of water and primary alcohol,” J. Chem. Phys. 90, 3292-3294 (1989).
[CrossRef]

1987

J. B. Hasted, S. K. Husain, F. A. M. Frescura, and J. R. Birch, “The temperature variation of the near millimetre wavelength optical constants of water,” Infrared Phys. 27, 11-15 (1987).
[CrossRef]

1985

R. M. Hill and L. A. Dissado, “Debye and non-Debye relaxation,” J. Phys. C 18, 3829-3836 (1985).
[CrossRef]

1977

1976

A. K. Lyaschenko, V. S. Goncharov, and P. S. Yastremskii, “Structure and dielectric properties of aqueous solutions of hydrogen peroxide,” J. Struct. Chem. 17, 871-876 (1976).
[CrossRef]

1975

M. T. Tyn and W. F. Calus, “Temperature and concentration dependence of mutual diffusion coefficients of some binary liquid systems,” J. Chem. Eng. Data 20, 310-316 (1975).
[CrossRef]

1971

V. V. Levin, E. A. Ovsoyan, and V. I. Ovchinnikov, “Dispersion of the dielectric constant of nitromethane,” Russian J. Phys. Chem. 45, 1333-1334 (1971).

1967

J. A. Boyne and A. G. Williamson, “Enthalpies of mixture of ethanol and water at 25°C,” J. Chem. Eng. Data 12, 318 (1967).
[CrossRef]

1950

P. M. Gross, Jr. and R. C. Taylor, “The dielectric constants of water, hydrogen peroxide, and hydrogen peroxide-water mixtures,” J. Am. Chem. Soc. 72, 2075-2080 (1950).
[CrossRef]

Abu-Awwad, F.

H. E. Alper, F. Abu-Awwad, and P. Politzer, “Molecular dynamics simulations of liquid nitromethane,” J. Phys. Chem. B 103, 9738-9742 (1999).
[CrossRef]

Afsar, M. N.

Alison, J. M.

J. M. Alison and R. J. Sheppard, “A precision waveguide system for the measurement of complex permittivity of lossy liquids and solid tissues in the frequency range 29 GHzto90 GHz--III. The liquid system for 57to82GHz: an investigation into water and formamide,” Meas. Sci. Technol. 2, 975-979 (1991).
[CrossRef]

Alper, H. E.

H. E. Alper, F. Abu-Awwad, and P. Politzer, “Molecular dynamics simulations of liquid nitromethane,” J. Phys. Chem. B 103, 9738-9742 (1999).
[CrossRef]

Arikawa, T.

T. Arikawa, M. Nagai, and K. Tanaka, “Characterizing hydration state in solution using terahertz time-domain attenuated total reflection spectroscopy,” Chem. Phys. Lett. 457, 12-17 (2008).
[CrossRef]

M. Nagai, H. Yada, T. Arikawa, and K. Tanaka, “Terahertz time-domain attenuated total reflection spectroscopy in water and biological solution,” Int. J. Infrared Millim. Waves 27, 505-515 (2006).
[CrossRef]

Astrand, P.-O.

C. Rønne, L. Thrane, P.-O. Astrand, A. Wallqvist, K. V. Mikkelsen, and S. R. Keiding, “Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation,” J. Chem. Phys. 107, 5319-5331 (1997).
[CrossRef]

Bachhuber, K.

J. Barthel, K. Bachhuber, R. Buchner, and H. Hetzenauer, “Dielectric spectra of some common solvents in the microwave region. Water and lower alcohols,” Chem. Phys. Lett. 165, 369-373 (1990).
[CrossRef]

Barthel, J.

J. Barthel, K. Bachhuber, R. Buchner, and H. Hetzenauer, “Dielectric spectra of some common solvents in the microwave region. Water and lower alcohols,” Chem. Phys. Lett. 165, 369-373 (1990).
[CrossRef]

Birch, J. R.

J. B. Hasted, S. K. Husain, F. A. M. Frescura, and J. R. Birch, “The temperature variation of the near millimetre wavelength optical constants of water,” Infrared Phys. 27, 11-15 (1987).
[CrossRef]

Boyd, J. E.

J. E. Boyd, A. Briskman, C. M. Sayes, D. Mittleman, and V. Colvin, “Terahertz vibrational modes of inverse micelles,” J. Phys. Chem. B 106, 6346-6353 (2002).
[CrossRef]

J. E. Boyd, A. Briskman, V. L. Colvin, and D. M. Mittleman, “Direct observation of terahertz surface modes in nanometer-sized liquid water pools,” Phys. Rev. Lett. 87, 147401 (2001).
[CrossRef] [PubMed]

Boyne, J. A.

J. A. Boyne and A. G. Williamson, “Enthalpies of mixture of ethanol and water at 25°C,” J. Chem. Eng. Data 12, 318 (1967).
[CrossRef]

Briskman, A.

J. E. Boyd, A. Briskman, C. M. Sayes, D. Mittleman, and V. Colvin, “Terahertz vibrational modes of inverse micelles,” J. Phys. Chem. B 106, 6346-6353 (2002).
[CrossRef]

J. E. Boyd, A. Briskman, V. L. Colvin, and D. M. Mittleman, “Direct observation of terahertz surface modes in nanometer-sized liquid water pools,” Phys. Rev. Lett. 87, 147401 (2001).
[CrossRef] [PubMed]

Bründermann, E.

M. Heyden, E. Bründermann, U. Heugen, G. Niehues, D. M. Leitner, and M. Havenith, “Long-range influence of carbohydrates on the solvation dynamics of water--Answers from terahertz absorption measurements and molecular modeling simulations,” J. Am. Chem. Soc. 130, 5773-5779 (2008).
[CrossRef] [PubMed]

U. Heugen, G. Schwaab, E. Bründermann, M. Heyden, X. Yu, D. M. Leitner, and M. Havenith, “Solute-induced retardation of water dynamics probed directly by terahertz spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 12301-12306 (2006).
[CrossRef] [PubMed]

Buchner, R.

S. Schrödle, B. Fischer, H. Helm, and R. Buchner, “Picosecond dynamics and microheterogenity of water+dioxane mixtures,” J. Phys. Chem. A 111, 2043-2046 (2007).
[CrossRef] [PubMed]

T. Sato and R. Buchner, “Dielectric relaxation processes in ethanol/water mixtures,” J. Phys. Chem. A 108, 5007-5015 (2004).
[CrossRef]

J. Barthel, K. Bachhuber, R. Buchner, and H. Hetzenauer, “Dielectric spectra of some common solvents in the microwave region. Water and lower alcohols,” Chem. Phys. Lett. 165, 369-373 (1990).
[CrossRef]

Calus, W. F.

M. T. Tyn and W. F. Calus, “Temperature and concentration dependence of mutual diffusion coefficients of some binary liquid systems,” J. Chem. Eng. Data 20, 310-316 (1975).
[CrossRef]

Chang, R.

R. Chang, Physical Chemistry for the Chemical and Biological Sciences, 3rd ed. (University Science Books, 2000).

Chiba, A.

T. Sato, A. Chiba, and R. Nozaki, “Dynamical aspects of mixing schemes in ethanol-water mixtures in terms of the excess partial molar activation free energy, enthalpy, and entropy of the dielectric relaxation process,” J. Chem. Phys. 110, 2508-2521 (1999).
[CrossRef]

Colvin, V.

J. E. Boyd, A. Briskman, C. M. Sayes, D. Mittleman, and V. Colvin, “Terahertz vibrational modes of inverse micelles,” J. Phys. Chem. B 106, 6346-6353 (2002).
[CrossRef]

Colvin, V. L.

J. E. Boyd, A. Briskman, V. L. Colvin, and D. M. Mittleman, “Direct observation of terahertz surface modes in nanometer-sized liquid water pools,” Phys. Rev. Lett. 87, 147401 (2001).
[CrossRef] [PubMed]

D. M. Mittleman, M. C. Nuss, and V. L. Colvin, “Terahertz spectroscopy of water in inverse micelles,” Chem. Phys. Lett. 275, 332-338 (1997).
[CrossRef]

Davis, M.

N. E. Hill, W. E. Vaughan, A. H. Price, and M. Davis, Dielectric Properties and Molecular Behaviour (Van Nostrand Reinhold Company, 1969).

Dissado, L. A.

R. M. Hill and L. A. Dissado, “Debye and non-Debye relaxation,” J. Phys. C 18, 3829-3836 (1985).
[CrossRef]

Fattinger, Ch.

Fischer, B.

S. Schrödle, B. Fischer, H. Helm, and R. Buchner, “Picosecond dynamics and microheterogenity of water+dioxane mixtures,” J. Phys. Chem. A 111, 2043-2046 (2007).
[CrossRef] [PubMed]

Fischer, B. M.

Frescura, F. A. M.

J. B. Hasted, S. K. Husain, F. A. M. Frescura, and J. R. Birch, “The temperature variation of the near millimetre wavelength optical constants of water,” Infrared Phys. 27, 11-15 (1987).
[CrossRef]

Goncharov, V. S.

A. K. Lyaschenko, V. S. Goncharov, and P. S. Yastremskii, “Structure and dielectric properties of aqueous solutions of hydrogen peroxide,” J. Struct. Chem. 17, 871-876 (1976).
[CrossRef]

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Gross, P. M.

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J. B. Hasted, S. K. Husain, F. A. M. Frescura, and J. R. Birch, “The temperature variation of the near millimetre wavelength optical constants of water,” Infrared Phys. 27, 11-15 (1987).
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M. Heyden, E. Bründermann, U. Heugen, G. Niehues, D. M. Leitner, and M. Havenith, “Long-range influence of carbohydrates on the solvation dynamics of water--Answers from terahertz absorption measurements and molecular modeling simulations,” J. Am. Chem. Soc. 130, 5773-5779 (2008).
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U. Heugen, G. Schwaab, E. Bründermann, M. Heyden, X. Yu, D. M. Leitner, and M. Havenith, “Solute-induced retardation of water dynamics probed directly by terahertz spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 12301-12306 (2006).
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A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321-2323 (1996).
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S. Schrödle, B. Fischer, H. Helm, and R. Buchner, “Picosecond dynamics and microheterogenity of water+dioxane mixtures,” J. Phys. Chem. A 111, 2043-2046 (2007).
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J. Barthel, K. Bachhuber, R. Buchner, and H. Hetzenauer, “Dielectric spectra of some common solvents in the microwave region. Water and lower alcohols,” Chem. Phys. Lett. 165, 369-373 (1990).
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Heugen, U.

M. Heyden, E. Bründermann, U. Heugen, G. Niehues, D. M. Leitner, and M. Havenith, “Long-range influence of carbohydrates on the solvation dynamics of water--Answers from terahertz absorption measurements and molecular modeling simulations,” J. Am. Chem. Soc. 130, 5773-5779 (2008).
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U. Heugen, G. Schwaab, E. Bründermann, M. Heyden, X. Yu, D. M. Leitner, and M. Havenith, “Solute-induced retardation of water dynamics probed directly by terahertz spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 12301-12306 (2006).
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M. Heyden, E. Bründermann, U. Heugen, G. Niehues, D. M. Leitner, and M. Havenith, “Long-range influence of carbohydrates on the solvation dynamics of water--Answers from terahertz absorption measurements and molecular modeling simulations,” J. Am. Chem. Soc. 130, 5773-5779 (2008).
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U. Heugen, G. Schwaab, E. Bründermann, M. Heyden, X. Yu, D. M. Leitner, and M. Havenith, “Solute-induced retardation of water dynamics probed directly by terahertz spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 12301-12306 (2006).
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Husain, S. K.

J. B. Hasted, S. K. Husain, F. A. M. Frescura, and J. R. Birch, “The temperature variation of the near millimetre wavelength optical constants of water,” Infrared Phys. 27, 11-15 (1987).
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P. Uhd Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B 13, 2424-2436 (1996).
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L. Thrane, R. H. Jacobsen, P. Uhd Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330-333 (1995).
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Kaatze, U.

P. Petong, R. Pottel, and U. Kaatze, “Water-ethanol mixtures at different compositions and temperatures. A dieletric relaxation study,” J. Phys. Chem. A 104, 7420-7428 (2000).
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U. Kaatze, “Complex permittivity of water as a function of frequency and temperature,” J. Chem. Eng. Data 34, 371-374 (1989).
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Keiding, S. R.

C. Rønne, L. Thrane, P.-O. Astrand, A. Wallqvist, K. V. Mikkelsen, and S. R. Keiding, “Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation,” J. Chem. Phys. 107, 5319-5331 (1997).
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P. Uhd Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B 13, 2424-2436 (1996).
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L. Thrane, R. H. Jacobsen, P. Uhd Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330-333 (1995).
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J. T. Kindt and C. A. Schmuttenmaer, “Far-Infrared dielectric properties of polar liquids probed by femtosecond terahertz pulse spectroscopy,” J. Phys. Chem. 100, 373-379 (1996).
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H. Kitahara, T. Yagi, K. Mano, M. W. Takeda, S. Kojima, and S. Nishizawa, “Dielectric characteristics of water solutions of ethanol in the terahertz region,” J. Korean Phys. Soc. 46, 82-85 (2005).

Kojima, S.

H. Kitahara, T. Yagi, K. Mano, M. W. Takeda, S. Kojima, and S. Nishizawa, “Dielectric characteristics of water solutions of ethanol in the terahertz region,” J. Korean Phys. Soc. 46, 82-85 (2005).

Kuwabara, S.

S. Mashimo, S. Kuwabara, S. Yagihara, and K. Higasi, “The dielectric relaxation of mixtures of water and primary alcohol,” J. Chem. Phys. 90, 3292-3294 (1989).
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Leitner, D. M.

M. Heyden, E. Bründermann, U. Heugen, G. Niehues, D. M. Leitner, and M. Havenith, “Long-range influence of carbohydrates on the solvation dynamics of water--Answers from terahertz absorption measurements and molecular modeling simulations,” J. Am. Chem. Soc. 130, 5773-5779 (2008).
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U. Heugen, G. Schwaab, E. Bründermann, M. Heyden, X. Yu, D. M. Leitner, and M. Havenith, “Solute-induced retardation of water dynamics probed directly by terahertz spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 12301-12306 (2006).
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Lyaschenko, A. K.

A. K. Lyaschenko, V. S. Goncharov, and P. S. Yastremskii, “Structure and dielectric properties of aqueous solutions of hydrogen peroxide,” J. Struct. Chem. 17, 871-876 (1976).
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Mano, K.

H. Kitahara, T. Yagi, K. Mano, M. W. Takeda, S. Kojima, and S. Nishizawa, “Dielectric characteristics of water solutions of ethanol in the terahertz region,” J. Korean Phys. Soc. 46, 82-85 (2005).

Mashimo, S.

S. Mashimo, S. Kuwabara, S. Yagihara, and K. Higasi, “The dielectric relaxation of mixtures of water and primary alcohol,” J. Chem. Phys. 90, 3292-3294 (1989).
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P. Uhd Jepsen, U. Møller, and H. Merbold, “Investigation of aqueous alcohol and sugar solutions with reflection terahertz time-domain spectroscopy,” Opt. Express 15, 717-737 (2007).
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C. Rønne, L. Thrane, P.-O. Astrand, A. Wallqvist, K. V. Mikkelsen, and S. R. Keiding, “Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation,” J. Chem. Phys. 107, 5319-5331 (1997).
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Mittleman, D.

J. E. Boyd, A. Briskman, C. M. Sayes, D. Mittleman, and V. Colvin, “Terahertz vibrational modes of inverse micelles,” J. Phys. Chem. B 106, 6346-6353 (2002).
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Mittleman, D. M.

J. E. Boyd, A. Briskman, V. L. Colvin, and D. M. Mittleman, “Direct observation of terahertz surface modes in nanometer-sized liquid water pools,” Phys. Rev. Lett. 87, 147401 (2001).
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D. M. Mittleman, M. C. Nuss, and V. L. Colvin, “Terahertz spectroscopy of water in inverse micelles,” Chem. Phys. Lett. 275, 332-338 (1997).
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P. Uhd Jepsen, J. K. Nielsen, and U. Møller, “Characterization of aqueous alcohol solutions in bottles with THz reflection spectroscopy,” Opt. Express 16, 9318-9331 (2008).
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P. Uhd Jepsen, U. Møller, and H. Merbold, “Investigation of aqueous alcohol and sugar solutions with reflection terahertz time-domain spectroscopy,” Opt. Express 15, 717-737 (2007).
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Nagai, M.

H. Yada, M. Nagai, and K. Tanaka, “The intermolecular stretching vibration mode in water isotopes investigated with broadband terahertz time-domain spectroscopy,” Chem. Phys. Lett. 473, 279-283 (2009).
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T. Arikawa, M. Nagai, and K. Tanaka, “Characterizing hydration state in solution using terahertz time-domain attenuated total reflection spectroscopy,” Chem. Phys. Lett. 457, 12-17 (2008).
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H. Yada, M. Nagai, and K. Tanaka, “Origin of the fast relaxation component of water and heavy water revealed by terahertz time-domain attenuated total reflection spectroscopy,” Chem. Phys. Lett. 464, 166-170 (2008).
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M. Nagai, H. Yada, T. Arikawa, and K. Tanaka, “Terahertz time-domain attenuated total reflection spectroscopy in water and biological solution,” Int. J. Infrared Millim. Waves 27, 505-515 (2006).
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H. Hirori, M. Nagai, and K. Tanaka, “Destructive interference effect on surface plasmon resonance in terahertz attenuated total reflection,” Opt. Express 13, 10801-10814 (2005).
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A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321-2323 (1996).
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Niehues, G.

M. Heyden, E. Bründermann, U. Heugen, G. Niehues, D. M. Leitner, and M. Havenith, “Long-range influence of carbohydrates on the solvation dynamics of water--Answers from terahertz absorption measurements and molecular modeling simulations,” J. Am. Chem. Soc. 130, 5773-5779 (2008).
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Nielsen, J. K.

Nishizawa, S.

H. Kitahara, T. Yagi, K. Mano, M. W. Takeda, S. Kojima, and S. Nishizawa, “Dielectric characteristics of water solutions of ethanol in the terahertz region,” J. Korean Phys. Soc. 46, 82-85 (2005).

Nozaki, R.

T. Sato, A. Chiba, and R. Nozaki, “Dynamical aspects of mixing schemes in ethanol-water mixtures in terms of the excess partial molar activation free energy, enthalpy, and entropy of the dielectric relaxation process,” J. Chem. Phys. 110, 2508-2521 (1999).
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D. M. Mittleman, M. C. Nuss, and V. L. Colvin, “Terahertz spectroscopy of water in inverse micelles,” Chem. Phys. Lett. 275, 332-338 (1997).
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V. V. Levin, E. A. Ovsoyan, and V. I. Ovchinnikov, “Dispersion of the dielectric constant of nitromethane,” Russian J. Phys. Chem. 45, 1333-1334 (1971).

Ovsoyan, E. A.

V. V. Levin, E. A. Ovsoyan, and V. I. Ovchinnikov, “Dispersion of the dielectric constant of nitromethane,” Russian J. Phys. Chem. 45, 1333-1334 (1971).

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T. J. Parker, “Dispersive Fourier transform spectroscopy,” Contemp. Phys. 31, 335-353 (1990).
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P. Petong, R. Pottel, and U. Kaatze, “Water-ethanol mixtures at different compositions and temperatures. A dieletric relaxation study,” J. Phys. Chem. A 104, 7420-7428 (2000).
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P. Petong, R. Pottel, and U. Kaatze, “Water-ethanol mixtures at different compositions and temperatures. A dieletric relaxation study,” J. Phys. Chem. A 104, 7420-7428 (2000).
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N. E. Hill, W. E. Vaughan, A. H. Price, and M. Davis, Dielectric Properties and Molecular Behaviour (Van Nostrand Reinhold Company, 1969).

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D. C. Sorescu, B. M. Rice, and D. L. Thompson, “Molecular dynamics simulations of liquid nitromethane,” J. Phys. Chem. A 105, 9336-9346 (2001).
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C. Rønne, L. Thrane, P.-O. Astrand, A. Wallqvist, K. V. Mikkelsen, and S. R. Keiding, “Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation,” J. Chem. Phys. 107, 5319-5331 (1997).
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T. Sato and R. Buchner, “Dielectric relaxation processes in ethanol/water mixtures,” J. Phys. Chem. A 108, 5007-5015 (2004).
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T. Sato, A. Chiba, and R. Nozaki, “Dynamical aspects of mixing schemes in ethanol-water mixtures in terms of the excess partial molar activation free energy, enthalpy, and entropy of the dielectric relaxation process,” J. Chem. Phys. 110, 2508-2521 (1999).
[CrossRef]

Sayes, C. M.

J. E. Boyd, A. Briskman, C. M. Sayes, D. Mittleman, and V. Colvin, “Terahertz vibrational modes of inverse micelles,” J. Phys. Chem. B 106, 6346-6353 (2002).
[CrossRef]

Schmuttenmaer, C. A.

J. T. Kindt and C. A. Schmuttenmaer, “Far-Infrared dielectric properties of polar liquids probed by femtosecond terahertz pulse spectroscopy,” J. Phys. Chem. 100, 373-379 (1996).
[CrossRef]

Schrödle, S.

S. Schrödle, B. Fischer, H. Helm, and R. Buchner, “Picosecond dynamics and microheterogenity of water+dioxane mixtures,” J. Phys. Chem. A 111, 2043-2046 (2007).
[CrossRef] [PubMed]

Schwaab, G.

U. Heugen, G. Schwaab, E. Bründermann, M. Heyden, X. Yu, D. M. Leitner, and M. Havenith, “Solute-induced retardation of water dynamics probed directly by terahertz spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 12301-12306 (2006).
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J. M. Alison and R. J. Sheppard, “A precision waveguide system for the measurement of complex permittivity of lossy liquids and solid tissues in the frequency range 29 GHzto90 GHz--III. The liquid system for 57to82GHz: an investigation into water and formamide,” Meas. Sci. Technol. 2, 975-979 (1991).
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Sorescu, D. C.

D. C. Sorescu, B. M. Rice, and D. L. Thompson, “Molecular dynamics simulations of liquid nitromethane,” J. Phys. Chem. A 105, 9336-9346 (2001).
[CrossRef]

Takeda, M. W.

H. Kitahara, T. Yagi, K. Mano, M. W. Takeda, S. Kojima, and S. Nishizawa, “Dielectric characteristics of water solutions of ethanol in the terahertz region,” J. Korean Phys. Soc. 46, 82-85 (2005).

Tanaka, K.

H. Yada, M. Nagai, and K. Tanaka, “The intermolecular stretching vibration mode in water isotopes investigated with broadband terahertz time-domain spectroscopy,” Chem. Phys. Lett. 473, 279-283 (2009).
[CrossRef]

T. Arikawa, M. Nagai, and K. Tanaka, “Characterizing hydration state in solution using terahertz time-domain attenuated total reflection spectroscopy,” Chem. Phys. Lett. 457, 12-17 (2008).
[CrossRef]

H. Yada, M. Nagai, and K. Tanaka, “Origin of the fast relaxation component of water and heavy water revealed by terahertz time-domain attenuated total reflection spectroscopy,” Chem. Phys. Lett. 464, 166-170 (2008).
[CrossRef]

M. Nagai, H. Yada, T. Arikawa, and K. Tanaka, “Terahertz time-domain attenuated total reflection spectroscopy in water and biological solution,” Int. J. Infrared Millim. Waves 27, 505-515 (2006).
[CrossRef]

H. Hirori, M. Nagai, and K. Tanaka, “Destructive interference effect on surface plasmon resonance in terahertz attenuated total reflection,” Opt. Express 13, 10801-10814 (2005).
[CrossRef] [PubMed]

Taylor, R. C.

P. M. Gross, Jr. and R. C. Taylor, “The dielectric constants of water, hydrogen peroxide, and hydrogen peroxide-water mixtures,” J. Am. Chem. Soc. 72, 2075-2080 (1950).
[CrossRef]

Thompson, D. L.

D. C. Sorescu, B. M. Rice, and D. L. Thompson, “Molecular dynamics simulations of liquid nitromethane,” J. Phys. Chem. A 105, 9336-9346 (2001).
[CrossRef]

Thrane, L.

C. Rønne, L. Thrane, P.-O. Astrand, A. Wallqvist, K. V. Mikkelsen, and S. R. Keiding, “Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation,” J. Chem. Phys. 107, 5319-5331 (1997).
[CrossRef]

L. Thrane, R. H. Jacobsen, P. Uhd Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330-333 (1995).
[CrossRef]

Tyn, M. T.

M. T. Tyn and W. F. Calus, “Temperature and concentration dependence of mutual diffusion coefficients of some binary liquid systems,” J. Chem. Eng. Data 20, 310-316 (1975).
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Vaughan, W. E.

N. E. Hill, W. E. Vaughan, A. H. Price, and M. Davis, Dielectric Properties and Molecular Behaviour (Van Nostrand Reinhold Company, 1969).

Wallqvist, A.

C. Rønne, L. Thrane, P.-O. Astrand, A. Wallqvist, K. V. Mikkelsen, and S. R. Keiding, “Investigation of the temperature dependence of dielectric relaxation in liquid water by THz reflection spectroscopy and molecular dynamics simulation,” J. Chem. Phys. 107, 5319-5331 (1997).
[CrossRef]

Weling, A. S.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321-2323 (1996).
[CrossRef]

Williamson, A. G.

J. A. Boyne and A. G. Williamson, “Enthalpies of mixture of ethanol and water at 25°C,” J. Chem. Eng. Data 12, 318 (1967).
[CrossRef]

Yada, H.

H. Yada, M. Nagai, and K. Tanaka, “The intermolecular stretching vibration mode in water isotopes investigated with broadband terahertz time-domain spectroscopy,” Chem. Phys. Lett. 473, 279-283 (2009).
[CrossRef]

H. Yada, M. Nagai, and K. Tanaka, “Origin of the fast relaxation component of water and heavy water revealed by terahertz time-domain attenuated total reflection spectroscopy,” Chem. Phys. Lett. 464, 166-170 (2008).
[CrossRef]

M. Nagai, H. Yada, T. Arikawa, and K. Tanaka, “Terahertz time-domain attenuated total reflection spectroscopy in water and biological solution,” Int. J. Infrared Millim. Waves 27, 505-515 (2006).
[CrossRef]

Yagi, T.

H. Kitahara, T. Yagi, K. Mano, M. W. Takeda, S. Kojima, and S. Nishizawa, “Dielectric characteristics of water solutions of ethanol in the terahertz region,” J. Korean Phys. Soc. 46, 82-85 (2005).

Yagihara, S.

S. Mashimo, S. Kuwabara, S. Yagihara, and K. Higasi, “The dielectric relaxation of mixtures of water and primary alcohol,” J. Chem. Phys. 90, 3292-3294 (1989).
[CrossRef]

Yang, X.

C. Zhang and X. Yang, “Molecular dynamics simulation of ethanol/water mixtures for structure and diffusion properties,” Fluid Phase Equilib. 231, 1-10 (2005).
[CrossRef]

Yastremskii, P. S.

A. K. Lyaschenko, V. S. Goncharov, and P. S. Yastremskii, “Structure and dielectric properties of aqueous solutions of hydrogen peroxide,” J. Struct. Chem. 17, 871-876 (1976).
[CrossRef]

Yu, X.

U. Heugen, G. Schwaab, E. Bründermann, M. Heyden, X. Yu, D. M. Leitner, and M. Havenith, “Solute-induced retardation of water dynamics probed directly by terahertz spectroscopy,” Proc. Natl. Acad. Sci. U.S.A. 103, 12301-12306 (2006).
[CrossRef] [PubMed]

Zhang, C.

C. Zhang and X. Yang, “Molecular dynamics simulation of ethanol/water mixtures for structure and diffusion properties,” Fluid Phase Equilib. 231, 1-10 (2005).
[CrossRef]

Zhang, J.

Appl. Phys. Lett.

A. Nahata, A. S. Weling, and T. F. Heinz, “A wideband coherent terahertz spectroscopy system using optical rectification and electro-optic sampling,” Appl. Phys. Lett. 69, 2321-2323 (1996).
[CrossRef]

Chem. Phys. Lett.

L. Thrane, R. H. Jacobsen, P. Uhd Jepsen, and S. R. Keiding, “THz reflection spectroscopy of liquid water,” Chem. Phys. Lett. 240, 330-333 (1995).
[CrossRef]

H. Yada, M. Nagai, and K. Tanaka, “Origin of the fast relaxation component of water and heavy water revealed by terahertz time-domain attenuated total reflection spectroscopy,” Chem. Phys. Lett. 464, 166-170 (2008).
[CrossRef]

T. Arikawa, M. Nagai, and K. Tanaka, “Characterizing hydration state in solution using terahertz time-domain attenuated total reflection spectroscopy,” Chem. Phys. Lett. 457, 12-17 (2008).
[CrossRef]

D. M. Mittleman, M. C. Nuss, and V. L. Colvin, “Terahertz spectroscopy of water in inverse micelles,” Chem. Phys. Lett. 275, 332-338 (1997).
[CrossRef]

J. Barthel, K. Bachhuber, R. Buchner, and H. Hetzenauer, “Dielectric spectra of some common solvents in the microwave region. Water and lower alcohols,” Chem. Phys. Lett. 165, 369-373 (1990).
[CrossRef]

H. Yada, M. Nagai, and K. Tanaka, “The intermolecular stretching vibration mode in water isotopes investigated with broadband terahertz time-domain spectroscopy,” Chem. Phys. Lett. 473, 279-283 (2009).
[CrossRef]

Contemp. Phys.

T. J. Parker, “Dispersive Fourier transform spectroscopy,” Contemp. Phys. 31, 335-353 (1990).
[CrossRef]

Fluid Phase Equilib.

C. Zhang and X. Yang, “Molecular dynamics simulation of ethanol/water mixtures for structure and diffusion properties,” Fluid Phase Equilib. 231, 1-10 (2005).
[CrossRef]

Infrared Phys.

J. B. Hasted, S. K. Husain, F. A. M. Frescura, and J. R. Birch, “The temperature variation of the near millimetre wavelength optical constants of water,” Infrared Phys. 27, 11-15 (1987).
[CrossRef]

Int. J. Infrared Millim. Waves

M. Nagai, H. Yada, T. Arikawa, and K. Tanaka, “Terahertz time-domain attenuated total reflection spectroscopy in water and biological solution,” Int. J. Infrared Millim. Waves 27, 505-515 (2006).
[CrossRef]

J. Am. Chem. Soc.

M. Heyden, E. Bründermann, U. Heugen, G. Niehues, D. M. Leitner, and M. Havenith, “Long-range influence of carbohydrates on the solvation dynamics of water--Answers from terahertz absorption measurements and molecular modeling simulations,” J. Am. Chem. Soc. 130, 5773-5779 (2008).
[CrossRef] [PubMed]

P. M. Gross, Jr. and R. C. Taylor, “The dielectric constants of water, hydrogen peroxide, and hydrogen peroxide-water mixtures,” J. Am. Chem. Soc. 72, 2075-2080 (1950).
[CrossRef]

J. Chem. Eng. Data

M. T. Tyn and W. F. Calus, “Temperature and concentration dependence of mutual diffusion coefficients of some binary liquid systems,” J. Chem. Eng. Data 20, 310-316 (1975).
[CrossRef]

U. Kaatze, “Complex permittivity of water as a function of frequency and temperature,” J. Chem. Eng. Data 34, 371-374 (1989).
[CrossRef]

J. A. Boyne and A. G. Williamson, “Enthalpies of mixture of ethanol and water at 25°C,” J. Chem. Eng. Data 12, 318 (1967).
[CrossRef]

J. Chem. Phys.

S. Mashimo, S. Kuwabara, S. Yagihara, and K. Higasi, “The dielectric relaxation of mixtures of water and primary alcohol,” J. Chem. Phys. 90, 3292-3294 (1989).
[CrossRef]

T. Sato, A. Chiba, and R. Nozaki, “Dynamical aspects of mixing schemes in ethanol-water mixtures in terms of the excess partial molar activation free energy, enthalpy, and entropy of the dielectric relaxation process,” J. Chem. Phys. 110, 2508-2521 (1999).
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Absorption spectrum (red curve, logarithmic scale) and index of refraction (black curve, linear scale) as function of frequency (lower horizontal scale) and wavenumber (upper, horizontal scale). The gray area indicates the 0.1 2.5 THz region, and the narrow yellow shaded area indicates the visible part of the spectrum.

Fig. 2
Fig. 2

Experimental configurations for (a) dispersive Fourier transform spectroscopy (DFTS), (b) normal-incidence reflection THz time-domain spectroscopy (THz-TDS), (c) generalized reflection THz-TDS with an incidence angle θ, and (d) attenuated total reflection (ATR) THz-TDS with an internal angle θ larger than the total internal reflection angle.

Fig. 3
Fig. 3

(a) Real and (b) imaginary part of the complex dielectric function of 10 different ethanol–water mixtures at 298 K ( 25 ° C ) .

Fig. 4
Fig. 4

Real (solid symbols) and imaginary (open symbols) part of the complex dielectric function of (a) 11% and (b) 50% ethanol–water mixtures at 298 K (25°). Data recorded in the present work are shown with gray squares (green online), and data from Sato and Buchner [24] are shown in black squares. The fitted dielectric function is shown by solid (real part) and dashed (imaginary part) curves, respectively. The contributions to ϵ from the three relaxation processes and the vibrational term are shown in shades of gray (blue online).

Fig. 5
Fig. 5

Fitted curves of (a) the real and (b) the imaginary part of the complex dielectric functions for different ethanol–water mixtures by using Eq. (14).

Fig. 6
Fig. 6

Fitted values and standard deviations of (a) the static dielectric constant, ϵ s , and (b) the dielectric constant in the high frequency limit, ϵ , of ethanol–water mixtures as a function of ethanol fraction X Et O H .

Fig. 7
Fig. 7

Debye relaxation strengths (a) Δ ϵ 1 , (b) Δ ϵ 2 , and Δ ϵ 3 , normalized to the full dielectric strength ϵ s ϵ , of ethanol–water mixtures as functions of the ethanol concentration.

Fig. 8
Fig. 8

Mixing enthalpy of ethanol/water mixtures (red curve, from [39]) and the relaxation strengths of the slow Debye process (black points) as a function of ethanol concentration

Fig. 9
Fig. 9

Debye relaxation times (a) τ 1 , (b) τ 2 , and (c) τ 3 of ethanol–water mixtures as a function of ethanol concentration.

Fig. 10
Fig. 10

Mixing volume of ethanol–water mixtures (solid, red curve) and the time constants τ 2 (black squares).

Fig. 11
Fig. 11

(a) Resonance frequency, (b) damping constant, and (c) vibration strength of the intermolecular stretching vibration of ethanol–water mixtures as a function of ethanol concentration.

Fig. 12
Fig. 12

Self-referenced THz reflection spectroscopy of liquid inside a PET bottle.

Fig. 13
Fig. 13

Reflected THz traces from a PET bottle containing water (lower trace), ethanol (middle trace), and dioxan (upper trace).

Fig. 14
Fig. 14

(a) Reflected THz traces from a PET surface in contact with water (blue trace) and a 30% hydrogen peroxide solution (red trace). (b) Real and imaginary part of the dielectric function of water (blue traces) and a 30% hydrogen peroxide solution (red traces). The error bars indicate the standard deviation between five consecutive scans.

Fig. 15
Fig. 15

Complex permittivity of (a) nitromethane and and (b) nitroethane. Solid gray (red online) symbols and solid black (blue online) symbols represent our THz measurements, open symbols are literature values [48, 49], solid lines are fits using the double Debye model to the available experimental data.

Tables (3)

Tables Icon

Table 1 Comparison of Relaxation Parameters of Water

Tables Icon

Table 2 Comparison of Relaxation Parameters of Ethanol

Tables Icon

Table 3 Relaxation Parameters for Nitromethane and Nitroethane

Equations (15)

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R exp ( i ϕ ) = F { I w l ( τ ) } F { I w a ( τ ) } = I ̂ w l ( ν ) I ̂ w a ( ν ) = r ̂ w l ( ν ) r w a ( ν ) = ( n ̂ l ( ν ) n w ( ν ) ) ( 1 + n w ( ν ) ) ( n ̂ l ( ν ) + n w ( ν ) ) ( 1 n w ( ν ) ) ,
n l = 1 R 2 ( 1 n w 1 + n w ) 2 1 + R 2 ( 1 n w 1 + n w ) 2 2 R cos ϕ ( 1 n w 1 + n w ) ,
α l = 4 π ν c 2 R sin ϕ ( 1 n w 1 + n w ) 1 + R 2 ( 1 n w 1 + n w ) 2 2 R cos ϕ ( 1 n w 1 + n w ) .
F { E s ( τ ) } F { E r ( τ ) } = E ̂ s ( ν ) E ̂ r ( ν ) = t a w t w a r a w exp ( i 4 π n w d w ν c ) n ̂ l n w n ̂ l + n w R exp ( i ϕ ) = E ̂ s ( ν ) E ̂ r ( ν ) r a w t a w t w a exp ( i 4 π n w d w ν c ) = n ̂ l n w n ̂ l + n w ,
n l = n w ( 1 R 2 ) 1 + R 2 + 2 R cos ϕ ,
α l = 4 π n w ν c 2 R sin ϕ 1 + R 2 + 2 R cos ϕ .
R exp ( i ϕ ) = F { E s ( τ ) } F { E r ( τ ) } = E ̂ s ( ν ) E ̂ r ( ν ) = r ̂ w l p r ̂ w a p r ̂ w l p = r ̂ w a p R exp ( i ϕ ) ,
ϵ ̂ l = B ± B 2 A B sin 2 2 θ 2 A cos 2 θ ,
d P 2 d t = P P 1 P 2 τ ,
P 2 ( t ) = ( P P 1 ) ( 1 exp ( t τ ) ) ,
d P 2 d t = ϵ 0 ( ϵ s ϵ ) τ E 0 exp ( i 2 π ν t ) P 2 t
P 2 ( t ) = ϵ 0 ( ϵ s ϵ ) 1 i 2 π ν τ E ( t ) ,
ϵ ̂ ( ν ) = ϵ ( ν ) + i ϵ ( ν ) = ϵ + ϵ s ϵ 1 i 2 π ν τ .
ϵ ̂ ( ω ) = j = 1 N Δ ϵ j 1 i ω τ j + j = 1 M A j ω j 2 ω 2 i ω γ j + ϵ ,
ϵ s ϵ = Δ ϵ 1 + Δ ϵ 2 + Δ ϵ 3 + A V ω V 2 = ( ϵ s ϵ 1 ) + ( ϵ 1 ϵ 2 ) + ( ϵ 2 A V ω V 2 ϵ ) + A V ω V 2 ,

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