Effective Debye relaxation models for binary solutions of polar liquids in the terahertz frequencies

Abstract

There are many effective media models that accurately describe the dielectric properties of mixtures. However, these models assume that the components are non-interacting. This assumption is not valid for solutions of polar liquids, resulting in significant deviations between the measured and theoretically predicted complex index of refraction of the mixtures. We present three effective media theories by expanding the well-known Debye relaxation model for solutions of polar liquids in the terahertz (THz) regime. The new effective media models proposed in this paper predict the individual relaxation Debye parameters based on the cooperative motion dynamics and self-associative properties of each mixture, and therefore explain the deviation of the dielectric functions of the solutions from the traditional effective media models. These models are verified through reflection measurements of four alcohol-water solutions acquired through THz time-domain spectroscopy (THz-TDS). Compared to the current mixed media models, the new effective Debye theorem predicts the dielectric properties of polar solutions more accurately, and has the potential to explain inter-species mixing schemes and interactions.

Graphical Abstract

1. Introduction

Liquid water plays an important role in many biological processes due to its unique properties. Despite its utility and ubiquitous nature, some of the dynamic properties of water are not well understood and therefore are subject of ongoing research efforts.^1–3 The short lifetimes of its intermolecular interactions produce a dynamic hydrogen bond network that can conform to foreign bodies and its ability to associate with ions and other polar solutes make it a versatile solvent.⁴ These properties assist with metabolic processes such as protein folding or membrane transport.⁴ These structural qualities often result in non-idealities in water-solute mixtures. Previous works in the microwave and THz frequencies report unexpected structural dynamics in methanol-water,⁵, ethanol-water,^6,7 1-propanol-water,⁸ and 2-propanol-water^9,10 solutions, as well as non-alcohol mixtures such as acetonitrile-water, acetone-water¹¹ and ion-water solutions.¹²

Contrary to polar liquid-water solutions, the effective dielectric properties of most mixtures are well predicted by existing mixing models, such as linear additivity, Maxwell-Garnett, Bottcher, Kirk-wood, and Bruggeman theorems,¹³ These approximations are developed based on compressibility, particle size, embeddedness, or shape of the materials. The physical foundations of the derivations in these models rest on the assumption that the components do not interact with one another.¹⁴ However, this is not true for mixtures of polar liquids, where multiple types of intermolecular interactions such as, but not limited to, hydrogen bonding, steric effects, or self-association, can restrict the motion of molecules and change its bulk properties.^15–18 In the case of water-solute mixtures, the coupling of inter- and intramolecular vibrations of water may potentially delocalize the vibrational modes of the solute,¹ further influencing the mixing properties of the solution.

The deviation between theory and measured dielectric values has been widely reported.^{15,16,19–24} In the THz regime, the divergence is often used to describe molecular interactions such as intersecting hydrogen bond networks,²⁴ the packing density of foreign molecules,¹⁵ or hydration dynamics.^15,16 In particular, the differences in the index of refraction and absorption spectra at the lower THz frequencies (0.15 – 0.75 THz) correspond to disturbances in the hydrogen bond network of water.¹¹ Significantly, these studies reported on the insufficiency of existing effective media models in explaining the dielectric function of the mixtures due to the complexity of the hydrogen bond networks.^11,15,16,24

Previous attempts to extract the molar fraction through the dielectric properties of alcohol solutions have been developed in the microwave frequencies.²⁵ Although this method was reliable for alcohol-alcohol mixtures, it was unable to resolve composition of alcohol-water solutions. Lou et al. proposed an effective Debye relaxation model that more accurately predicted the complex dielectric function of mixed solutions.²⁶ This method requires separately calculating the effective Debye relaxation time, static dielectric constant, and high-frequency dielectric constant. These parameters are then used in the Debye equation to predict the expected complex dielectric function, but it only relied on the slow relaxation events. However, fast relaxation processes, such as bond formation and degeneration, dominate in the THz regime, and therefore the method is not sufficient to describe molecular activity in higher frequencies.

In this work, we present three mixed solution rules, named the “Effective Debye Rules,” that theoretically predict the effective dielectric spectra of the mixtures of water and alcohol. We developed these Effective Debye Rules based on the cooperative motion between water and alcohol molecules. We extract the Debye relaxation parameters of pure water and alcohols. These values are then used in the proposed Effective Debye Rules to calculate three potential models for each solution. We verify this calculation by comparing the theoretical predictions to the frequency-dependent dielectric function of methanol-water, ethanol-water, 1-propanol-water, and 2-propanol-water solutions at different volume fractions. The models show a much-improved agreement with the experimental data as compared to the traditional predictive effective media theories (i.e. Bruggeman approximation). These Effective Debye Rules describe three distinct interaction mechanisms in different alcohol-water solutions, which imply the structure of the hydrogen bond network in the solutions.

2. Theory

In this section, we will describe the foundation for the derivations. First, we will describe the main equations that are common across all Effective Debye Rules. We highlight the key differences in the intermolecular behavior that are probed in the THz and microwave frequencies and the essential modifications made to expand the theory to capture faster dynamics. We will define the three Effective Debye Rules and explain the nuances between them that describe possible mixing dynamics.

2.1. The Effective Debye Model

In unperturbed conditions, polar molecules are randomly oriented within the medium. These molecules will reorient themselves with the direction of an applied external electric field. Upon the removal of the field, the molecules will relax back into an arbitrary position.^13,27 The polarizability of the medium is related to its permittivity via the Clausius-Mossotti equation, such that the microscopic characteristics define the macroscopic dielectric response of a polar liquid.^13,28 The frequency-dependent, complex dielectric spectra of polar molecules, $\tilde{\epsilon }(\omega )={\epsilon }^{\prime }(\omega )+j{\epsilon }^{″}(\omega )$, can be modeled by the Debye equation,²⁸

$$\tilde{\epsilon }(\omega )={\epsilon }_{\infty }+\frac{{\epsilon }_s-{\epsilon }_{\infty }}{1+j\omega \tau },$$ (1)

where ε_s, is the static dielectric constant, ε_∞ is the high-frequency constant, τ is the relaxation time, and ω is the angular frequency. This relaxation process for liquid water and alcohol is relatively slow, and has time constants range between 8 – 316 ps,²¹ corresponding to resonant frequencies between 3 – 100 GHz. In contrast to the microwave frequencies, the THz regime includes the fast relaxation processes related to the breaking and formation of hydrogen bonds, and each intermediate process has its own permittivity strength and time constant. Therefore, it is necessary to use multiple parameters to accurately describe broadband relaxation processes;^20,22,29 the expanded Debye equation for N relaxations is then,

$$\tilde{\epsilon }(\omega )={\epsilon }_{\infty }+\sum _{i=1}^N\frac{{\epsilon }_i-{\epsilon }_{i+1}}{1+j\omega {\tau }_i},$$ (2)

where ε₁ = ε_s, ε_N+1 = ε_∞, and ε_i and τ_i for 2 ≤ i ≤ N are the intermediate dielectric constants and relaxation times, respectively. The amount of relaxation events is related to the mesoscopic structure of the medium,³⁰ and occur only when specific “wait-and-switch” criteria are fulfilled.^31,32 The wait-and-switch process refers to the time a molecule waits for an opportunity to reorient itself, such as the introduction of a neighboring water molecule with a more favorable (i.e. lower) energy state. The molecule then switches its hydrogen bonds to form new bonds with its new neighbor.

Due to its ability to form four hydrogen bonds, water creates a wide-spanning hydrogen bond network.⁴ Thus, there are more surrounding hydrogen-bonding options available to a single water molecule compared to pure alcohols. The accessibility to its neighbors allows for a water molecule within the network to meet wait-and-switch criteria more readily. Additionally, monomers and dimers at the edges of one network can easily form hydrogen bonds to another network within the system. This event occurs at the same rate as the formation and degradation of hydrogen bonds within the network. These two motions are indistinguishable and result in an effective single relaxation.²⁹ Thus, liquid water ultimately has two relaxation events: a slow rotation, and a fast, translational motion related to the switching of hydrogen bonds between adjacent molecules.

On the contrary, alcohols form chained and branched hydrogen bond networks.^29–31 This results in a slower rotational speed for most alcohols in comparison to water, since the availability of neighboring molecules and networks is sparser. Like liquid water, an alcohol molecule can also switch hydrogen bonds with its neighbors. However, the molecules at the end of a chain or branch are less likely to meet its wait-and-switch criteria than those within the chain²⁹, resulting in two distinct resonances related to translational hydrogen bond switching. Therefore, liquid alcohol has three relaxation events: a slow rotational event, an intermediate hydrogen bonding event at the ends of the chain, and a fast event corresponding to the switching of hydrogen bonding events within the chain.

In a binary solution of water and alcohol, two polar liquids having double and triple Debye dielectric models are mixed. However, due to the differences in the expanded Debye models, additional Effective Debye Rules must be applied to the original theory proposed in Ref. 26. These Rules derive the effective dielectric and time constants from thermodynamic and electrochemical theory, while considering the intermolecular dynamics of the solutions, such that the Debye parameters corresponding to cooperative motions are mixed.

2.2. The common Effective Debye relations

In the following sections, we describe three common relationships for calculating the effective relaxation time, dielectric strength and high-frequency dielectric constant of solutions. These relationships are shared among the three Effective Debye Rules, which will be described in §2.3.

2.2.1. The effective relaxation times.

The relaxation processes require specific activation energy, ΔG^‡, in order to reorient the molecules. The rate of reaction, k, and energy required are related via the Eyring formula,³³

$$k=\frac{k_{B}T}{h}\text{exp }\left(\frac{-\Delta G^{‡}}{RT}\right),$$ (3)

where k_B is the Boltzmann’s constant, R is the molar gas constant, T is the temperature, and h is the Planck’s constant. The rate of reaction is the inverse of the relaxation time, and thus Eq. (3) can be rewritten as

$$\tau =\frac{h}{k_{B}T}\text{exp }\left(\frac{\Delta G^{‡}}{RT}\right).$$ (4)

For an ideally mixed solution, such that the structure is maintained and the energy is conserved, the total activation energy of the mixed states is linearly proportional to molar fraction of each solvent, or

$$\Delta G_{eff}^{‡}=x_A\Delta G_A^{‡}+x_W\Delta G_W^{‡},$$ (5)

where x is the molar fraction, and the subscripts A and W hereafter refer to alcohol and water, respectively, and the subscript ef f refers to the effective or mixed solution. From Eq.(4), ΔG^‡ ∝ ln τ, such that,

$$\Delta G^{‡}=RT\text{ ln }\left(\frac{k_{B}T\tau }{h}\right).$$ (6)

Substituting Eq. (6) in Eq.(5) results in,

$$RT\left(\text{ln }\left({\tau }_{eff}\right)+\text{ln }\left(\frac{k_{B}T}{h}\right)\right)=RT(x_A\left(\text{ln }\left({\tau }_A\right)+\text{ln }\left(\frac{k_{B}T}{h}\right)\right)\cdots +x_W\left(\text{ln }\left({\tau }_W\right)+\text{ln }\left(\frac{k_{B}T}{h}\right)\right)).$$ (7)

Given that here x_A +x_W = 1, Eq. (7) simplifies to

$$\text{ln }\left({\tau }_{eff}\right)=x_A\text{ ln }\left({\tau }_A\right)+x_W\text{ln }\left({\tau }_W\right).$$ (8)

We empirically find that using volume fraction, v, instead of molar fractions more accurately estimated the effective properties, and therefore we modified the relationship by,

$$\text{ln }\left({\tau }_{eff}\right)=v_A\text{ ln }\left({\tau }_A\right)+v_W\text{ ln }\left({\tau }_W\right).$$ (9)

The rotational processes of alcohol and water create a mixed-species hydrogen bond network that cooperatively rotate, and therefore τ_1,A and τ_1,W can be used in Eq. (9) to calculate τ_{1,ef f}. However, as previously stated, water and alcohol have two or three Debye relaxation responses, and the faster events dominate the dielectric spectra in the THz frequencies. Therefore, it is necessary to calculate the effective relaxation times for all processes. In order to account for various mixing schemes related to high frequency dynamics, additional τ_{ef f} relations are required, which are further explained in §2.3.

2.2.2. The dielectric constants.

The calculation of effective dielectric constants has been widely studied.¹⁴ Ref. 26 used the asymmetric oil-in-water Bruggeman model, in which alcohol is dispersed in a continuum of water. However, there are critical volumes in which the aqueous alcohol solutions switch roles from an oil-in-water medium to water-in-oil.³⁴ Therefore, in contrast to Ref. 26, we use the symmetric Bruggeman equation, which makes no assumption of the host medium. The i-th dielectric constant, ε_i,eff, can be modeled by,

$$\frac{v_W\left({\epsilon }_{i,W}-{\epsilon }_{i,eff}\right)}{{\epsilon }_{i,W}+2{\epsilon }_{i,eff}}+\frac{v_A\left({\epsilon }_{i,A}-{\epsilon }_{i,eff}\right)}{{\epsilon }_{i,A}+2{\epsilon }_{i,eff}}=0,$$ (10)

using the ε_i,W and ε_i,A of pure water and alcohol for 1 ≤ i ≤ N in Eq. (2).

2.2.3. The high frequency dielectric constant.

In impedance spectroscopy, the dielectric strengths are often modeled through circuit analogy.^13,35 In the Debye model, ε_∞ can be represented by a simple capacitor.¹³ For a pure substance, ε_∞ can be treated as a dielectric material between two parallel conducting plates, while in mixed media samples are represented as stacks of multiple dielectric material between the plates. The equivalent capacitance can be modeled as a parallel or series combination of the materials, depending on the configuration of the stacks. We follow the same assumption as in Ref. 26 and approximate ε_∞ by capacitors in series. Thus, ε_∞,eff for a binary mixture is given by,

$$\frac{1}{{\epsilon }_{\infty ,eff}}=\frac{v_A}{{\epsilon }_{\infty ,A}}+\frac{v_W}{{\epsilon }_{\infty ,W}}.$$ (11)

Eq. (11) will be used as a common relationship in the following three Effective Debye Rules.

2.3. The Effective Debye Rules

Multiple relaxation terms are needed to model pure water and alcohol in the THz regime. However, water has two resonances while alcohol has three. Therefore, the Debye parameters can not be combined using Eqs. (9) – (11) alone. Additional Effective Debye Rules are required to ensure all parameters are properly incorporated into the mixed media model.

2.3.1. Effective Debye Rule 1.

Both pure water and alcohol have relaxation modes that correspond to slow (i = 1) and fast resonances (i = N), where N = 2 for water and N = 3 for the alcohol. These cooperative Debye terms can be combined using Eqs. (9) and (10), i.e., the effective static dielectric constant of the mixture is calculated by mixing the static dielectric constants of water and alcohol using,

$$\frac{v_W\left({\epsilon }_{s,W}-{\epsilon }_{s,eff}\right)}{{\epsilon }_{s,W}+2{\epsilon }_{s,eff}}+\frac{v_A\left({\epsilon }_{s,A}-{\epsilon }_{s,eff}\right)}{{\epsilon }_{s,A}+2{\epsilon }_{s,eff}}=0,$$ (12)

and the ε_3,eff is calculated by mixing ε_3,A with ε_2,W using,

$$\frac{v_W\left({\epsilon }_{2,W}-{\epsilon }_{3,eff}\right)}{{\epsilon }_{2,W}+2{\epsilon }_{3,eff}}+\frac{v_A\left({\epsilon }_{3,A}-{\epsilon }_{3,eff}\right)}{{\epsilon }_{3,A}+2{\epsilon }_{3,eff}}=0.$$ (13)

Similarly, the effective slow and fast relaxation times are modeled by,

$$\text{ln }\left({\tau }_{1,eff}\right)=v_A\text{ ln }\left({\tau }_{1,A}\right)+v_W\text{ ln }\left({\tau }_{1,W}\right),$$ (14)

and

$$\text{ln }\left({\tau }_{3,eff}\right)=v_A\text{ ln }\left({\tau }_{3,A}\right)+v_W\text{ ln }\left({\tau }_{2,W}\right).$$ (15)

However, to calculate the intermediate Debye relaxation parameters of the solution, the differences between the alcohol and water hydrogen bond network structures should be considered carefully. In pure liquid alcohol, the molecules at the end of the chain-like hydrogen bond network are unaffected by the main structure. Therefore, the alcohol dielectric model exhibits an intermediate relaxation mode associated with the end-chain molecules motions. On the contrary, water does not have a corresponding intermediate relaxation. Therefore a separate Effective Debye Rule must be derived for the effective intermediate relaxation parameters, τ_2,eff and ε_2,eff.

It has been reported that the relaxation time of a pure solvent is inversely proportional to an applied hydrostatic pressure.^32,36 Increasing the pressure results in a higher density of molecules.³² This is analogous to increasing the volume fraction of a solvent in solution. As more cooperative molecules become available, the wait-and-switch criteria become more readily attainable. Thus, for the intermediate step associated with the end of the network of the alcohol, the effective relaxation time, τ_2,eff, at v_A > 0 can be estimated by,

$${\tau }_{2,eff}=\frac{{\tau }_{2,A}}{v_A}.$$ (16)

Furthermore, the strength of the intermediate relaxation of alcohol remains unchanged. Following the Claussius-Mossotti theorem, the solution’s polarizability is determined by the permittivity of its individual molecules and not affected by statistical events,²⁸ and therefore, for v_A > 0,

$${\epsilon }_{2,eff}={\epsilon }_{2,A}.$$ (17)

This Effective Debye Rule implies that the water and alcohol molecules create an inter-species hydrogen bond network. However, there are alcohol monomers that are excluded from the network structure. Equations (16) and (17) account for the relaxation dynamics of these molecules in the Effective Debye model.

2.3.2. Effective Debye Rule 2.

Here, we utilize the same relationship given by Eqs. (12) – (15) to calculate the static, slow and fast Debye relaxation parameters of the mixture. However, in contrast to Effective Debye Rule 1, the double Debye model of water is expanded to a triple Debye relationship to allow for mixing with alcohol’s intermediate relaxation mode. Liquid water’s large hydrogen bond network allows the molecules at the end of the network and within the network to switch bonds at the same rate.²⁹ Therefore, the second Effective Debye Rule assumes that the empirical parameters τ₂ and ε₂ of water’s double Debye model correspond to both the intermediate and fast relaxations of the pure alcohols. Therefore, we can expand the water’s dielectric model to a triple Debye equation, where

$${\tau }_{3,W}={\tau }_{2,W},$$ (18)

$${\epsilon }_{3,W}={\epsilon }_{2,W}.$$ (19)

This expansion allows alcohol and water to have an equal number of Debye parameters, which can then be mixed using Eqs. 9 and 10 to give the effective Debye parameters of the solution. At v_A = 0, ε₂,_{ef f} = ε₃,_eff = ε₂,_W. Therefore,

$${\tilde{\epsilon }}_W(\omega )={\epsilon }_{\infty ,W}+\frac{{\epsilon }_{s,W}-{\epsilon }_{2,W}}{1+j\omega {\tau }_{1,W}}+\frac{{\epsilon }_{2,W}-{\epsilon }_{2,W}}{1+j\omega {\tau }_{2,W}}+\frac{{\epsilon }_{2,W}-{\epsilon }_{\infty ,W}}{1+j\omega {\tau }_{2,W}}.$$ (20)

The intermediate term of Eq. (20) reduces to zero, and the triple Debye equation of water simplifies to a double relaxation model.

2.3.3. Effective Debye Rule 3.

The final Effective Debye Rule assumes that all alcohol and water molecules are in a collective and cooperative motion. In this Effective Debye Rule, the intermediate and fast relaxations of alcohol are indiscernible within the aqueous solution. Therefore, τ₂ represents the effective global relaxation of the solution.¹¹ Previous studies have shown the adequacy of the double Debye relationship for describing pure alcohol’s relaxation dynamics.^11,16 Thus, like water, we treat the intermediate and fast relaxations of alcohol as a single resonance, and reduce the triple Debye model for the alcohol to a two-relaxation relationship. Therefore, Eqs.(9) – (11) can now be used to give the effective Debye parameters of the solution.

The new Effective Debye Rules proposed here are summarized in Table 1. For ease of reference, we have named and abbreviated Eq. (9) to “MixTau,” Eq. (10) to “BGM,” and Eq. (11) to “MixInf.” The relaxation values extracted from 100% water and alcohols are used as inputs to Eqs. (9) – (20) to predict the Debye parameters of the mixed solutions. The process for extraction of the Debye parameters of the pure solvents is described in further detail in the methods section, below.

Table 1.

The three Effective Debye Rules proposed here to calculate the effective Debye model of alcohol and water solutions are summarized. In this table, Eq. (9) has been abbreviated to MixTau, Eq. (10) to BGM, and Eq. (11) to MixInf. The subscripts W and A refer to water and alcohol, respectively, and the subscript DD in Rule 3 refers to the double Debye parameters extracted from the pure alcohols.

	Rule 1	Rule 2	Rule 3 (Double Debye)
ε s	BGM(εs,W, εs,A)	BGM(εs,W, εs,A)	BGM(εs,W, εs,A,DD)
ε 2	ε 2 ,A	BGM(ε2,W, ε2,A)	BGM(ε2,W, ε2,A,DD)
ε 3	BGM(ε2,W, ε3,A)	BGM(ε2,W, ε3,A)	No third relaxation
ε ∞	MixInf(ε∞,W, ε∞,A)	MixInf(ε∞,W, ε∞,A)	MixInf(ε∞,W, ε∞,A,DD)
τ 1	MixTau(τ1,W, τ1,A)	MixTau(τ1,W, τ1,A)	MixTau(τ1,W, τ1,A,DD)
τ 2	τ2,A/vA	MixTau(τ2,W, τ2,A)	MixTau(τ2,W, τ2,A,DD)
τ 3	MixTau(τ2,W, τ3,A)	MixTau(τ2,W, τ3,A)	No third relaxation

3. Materials and Methods

3.1. Measurements

The three Rules used in the Effective Debye models were experimentally validated against the Bruggeman model in water-alcohol solutions mixed from 0 to 100 vol% alcohol at 20 vol% intervals. Methanol (MeOH), ethanol (EtOH), and 2-propanol (IPA) (≥ 99.9% purity, Thermo Fischer Scientific, Waltham, MA, USA), and 1-propanol (PrOH) (99% Sigma-Aldrich, St. Louis, MO, USA) were mixed with DI water with a resistivity of 18.2 MΩ·cm (Milli Q IQ7000, Millipore Sigma, St. Louis, MO, USA). A custom cuvette was constructed using HDPE for the side and back walls of the vessel. A 4-mm thick fused silica window (Edmund Optics, Barrington, NJ, USA) was adhered to the HDPE sides using a 2-part epoxy.

The measurement set up is illustrated in Fig. 1. We used a commercially-available THz spectrometer (TeraSmart, Menlo Systems GmbH, Martinsried, Germany) aligned in reflection geometry at normal incidence. A fiber-coupled InGaAs/InAlAs photoconductive antenna (PCA) emits a pulsed THz wave. The beam is collimated through a TPX-50 lens (Menlo Systems GmbH, Martinsried, Germany) and divided using a silicon beam-splitter (Si-BS) with a transmission/reflection ratio of 54/46 (Tydex, St. Petersburg, Russia). The transmitted light is focused and reflected from the quartz-sample interface of the cuvette, which is mounted on a motorized stage (Newport Corporation, Irvine, CA, USA). The cuvette is scanned and a row of 25, 1 mm pixels are measured; each pixel consists of 500 time averages. These spatial measurements are necessary to account for the fused silica window’s nonuniform thickness. This step is repeated for both reference (air), baseline (water), and sample measurements. Separate reflection measurements of water were used for the baseline correction algorithm.³⁷ The solutions were transferred into and out of the cuvette with a pipette, which was thoroughly cleaned between each sample. All measurements were acquired in a dry nitrogen-purged box (< 3% humidity) at 20±1°C.

Fig. 1.

The schematic of the optical set up is shown. A home-made cuvette was built using HDPE walls and a fused-silica window. The cuvette was mounted on a motorized linear stage. The THz beam was focused on the window-sample interface. The THz-TDS measurements were collected by translating the cuvette in 1 mm steps.

3.2. Data Analysis and Signal Processing

A material characterization algorithm was used to extract the samples’ complex index of refraction, $\tilde{n}(\omega )=n(\omega )-jk(\omega )$, where $\tilde{n}(\omega )=\sqrt{\tilde{\epsilon }(\omega )}$.^37,38 For pure water and alcohols, we follow the fitting procedure as described in Ref. 19, such that for a triple Debye model, ε_s, ε₂, and τ₁ were fixed to the values reported by Ref. 16. To extract the double Debye parameters of the pure alcohols, ε_s,DD is the approximate difference between ε_s and ε₃ extracted from the triple Debye model,¹⁶ and is held fixed.¹⁹ The Debye equations were fit to the measured dielectric properties of the pure solvents using a complex non-linear least squares optimization algorithm between 0.2 – 1.0 THz.

The extracted parameters, along with the published values, are shown in Table 2. The values extracted in this work are comparable to the previously reported relaxation parameters.^16,19,20,39 The relaxation parameters of the pure solvents are inputs for Eqs. (9) – (20). The Effective Debye Rules were used to calculate only the ${\epsilon }_{eff}^{\prime \prime }(\omega )$. The proposed models were able to predict the behavior of ${\epsilon }_{eff}^{\prime }(\omega )$ as a function of frequency, but not the exact values. To model both the real and imaginary parts of the dielectric function, we applied a singly-subtractive Kramers-Kronig (SSKK) transform to the predicted ${\epsilon }_{eff}^{\prime \prime }(\omega )$ between 0.2 – 1 THz, and anchored the predicted ${\epsilon }_{eff}^{\prime }(\omega )$ to ${\epsilon }_{\mathit{\text{meas}}}^{\prime }(\omega )$ at 0.5 THz.⁴⁰ We used a Kramers-Kronig software package freely available through MATLAB Central File Exchange.⁴¹

Table 2.

The Debye relaxation parameters, ε_s, ε_i, ε_∞, and τ_i, for pure (100%) methanol, ethanol, 1-propanol, 2-propanol, and water are compared to the reported values in Refs. 16, 19, 20, and 39. In the triple Debye models, the values for ε_s, ε₂, and τ₁ are set to the parameters reported in Ref. 16.

Model	Solute	Ref.	ε ∞	ε s	ε 2	ε 3	τ1 (ps)	τ2 (ps)	τ3 (fs)
Triple Debye	Methanol	This work	1.98	32.28	5.46	2.98	49.98	1.31	165
		(16)	1.79	32.28	5.46	3.00	49.98	1.22	107
		(20)	2.1	32.63	5.35	3.37	48	1.25	160
	Ethanol	This work	2.06	24.51	4.45	2.73	163.6	2.16	231
		(16)	2.00	24.51	4.45	3.03	163.6	3.98	222
		(19)	1.95	24.35	4.49	2.91	163	3.01	235
		(20)	1.93	24.35	4.15	2.72	161	3.3	220
	1-propanol	This work	2.11	20.3	3.61	2.5	320.5	2.82	265
		(16)	2.07	20.3	3.61	2.64	320.5	8.15	202
		(20)	1.97	20.44	3.43	2.37	316	2.9	200
	2-propanol	This work	2.07	19.42	3.65	2.55	350.9	4.24	256
		(16)	2.05	19.42	3.65	2.76	350.9	9.14	218
Double Debye	Water	This work	3.48	78.36		4.98	8.24		181
		(16)	3.09	78.36		5.11	7.39		108
		(19)	3.49	78.36		5.16	7.89		181
		(20)	3.48	78.36		4.93	8.24		180
		(39)	4.1	78.8		6.6	10.06		180
	Methanol	This work	2.12	29.3		4.52	12.09		252
		(16)	1.9	31.7		3.24	13.96		135
	Ethanol	This work	2.10	21.78		4.99	29.71		288
		(16)	1.98	24.5		2.75	19.23		155
	1-propanol	This work	2.12	17.8		2.5	44.09		319
		(16)	2.04	20.40		2.49	35.79		138
	2-propanol	This work	2.09	16.87		4.66	35.87		276
		(16)	2.01	19.43		2.53	28.45		139

4. Results

Figs. 2 – 5 show the measured complex index of refraction of alcohol-water solutions against the Bruggeman model and the optimal Effective Debye Rule over the entire spectrum. They also show the error of the effective media models averaged over 0.2–1.0 THz. The Effective Debye Rules 1, 2 and 3, each used as part of the proposed model, more accurately represent the dielectric response of mixed solutions of at least one of the alcohols and water compared to the Bruggeman model.

Fig. 2.

Compared to the Bruggeman model (red dashed lines), Rule 1 of the Effective Debye (black solid) are in excellent agreement with (a) the real and (b) the imaginary part of the index of refraction of methanol-water solutions. The color axis indicates the mixture percentage by volume. The average error of (c) n and (d) k illustrates that Effective Debye Rule 1 of the model shows the best improvement in comparison to the other Effective Debye Rules and the Bruggeman model.

Fig. 5.

Effective Debye Rule 3 best predicts the (a) n and (b) k of aqueous 2-propanol solutions, similar to 1-propanol-water mixtures. The agreement between different models is demonstrated in (c) and (d).

Methanol-water solutions best match with Effective Debye Rule 1 of the model. Figs. 2a and 2b show the complex indices of refraction of methanol solutions, the Bruggeman model, and Effective Debye Rule 1 of the proposed model. The red arrows in Figs. 2a and 2b illustrate increasing vol% of alcohol from 0 to 100%. In Fig. 2b, although it appears that there is agreement between the imaginary part of the index of refraction, k, as predicted by the Bruggeman model and the measurements, it should be noted that these values correspond to different vol% of alcohol. For instance, the 40% by volume modeled using the Bruggeman equation agrees with the measured extinction for the 20% by volume measurements. Figs. 2a and 2b show that the proposed Effective Debye Rule 1 predicts the complex index of refraction of methanol-water solutions more accurately in the lower frequency range between 0.2 to 0.9 THz, as compared to the higher frequencies above 0.9 THz. Figs. 2c and 2d better illustrate the error between the proposed Effective Debye Rules (Rules 1, 2, and 3), the Bruggeman model, and the extracted complex index of refraction. Here the difference between measured values and those predicted by the models are averaged between 0.2 and 1 THz, and plotted as a function of the volume percentage of the methanol. It can be seen that error of all models is negligible for pure samples at 0 and 100%, however the error increases significantly with higher solute percentage. After the 50% mid-point the role of the solute and solvent is reversed.

Fig. 3 shows that in ethanol-water solutions Effective Debye Rule 2 offers the most improved agreement with the extracted complex indices of refraction. However, it should be noted in Figs. 3c and 3d that the real part of the index of refraction is best described by Effective Debye Rule 1, whereas the imaginary part of the index of refraction follows Rule 2. Effective Debye Rule 1 (not shown) describes the low frequency behavior of ethanol-water solutions better while Effective Debye Rule 2 better captures the high-frequency effects.

Fig. 3.

Effective Debye Rule 2 best describes the complex index of refraction of ethanol solutions as illustrated in (a) and (b). The average error in (c) and (d) illustrates that there is a deviation between which rule best describes the n and k values. While n is best described by Rule 1, k is in better agreement with Rule 2.

Figs. 4 and 5 show similar spectroscopy results for aqueous 1-propanol and 2-propanol solutions, respectively. Here, Effective Debye Rule 3 offers the most accurate prediction of the complex index of refraction of aqueous 1-propanol and 2-propanol solutions, as compared to values predicted by the other Effective Debye Rules and the Bruggeman model. Because the propyl alcohols share similar hydrogen-bond network structures 42, it is expected that they follow the same Effective Debye Rule. In the following section, we will discuss the implication of agreement with specific Effective Debye Rules in understanding the complex hydrogen bond structure of different alcohol-water solutions.

Fig. 4.

Effective Debye Rule 3 best predicted the (a) real and (b) imaginary parts of the complex index of refraction of aqueous 1-propanol mixtures. As illustrated in (c) and (d) Effective Debye Rules 1 and 2 and the Bruggeman model pale in comparison to Rule 3.

5. Discussion

Non-idealities in the hydrogen-bond network of water manifest as deviations in the lower frequencies between measured values and traditional mixed media models.¹¹ The inadequacy of these traditional models in describing the dielectric constant of polar liquid solutions motivated our work to derive improved Effective Debye Rules. The three models derived in this paper account for different hydrogen-bond interactions and the cooperative motion between species.

For all aqueous alcohol solutions, there is a unique Effective Debye Rule that better fits the measured data than the Bruggeman model as well as the other Effective Debye Rules. This observation is clearly presented in parts c and d of Figs. 2, 3, 4, and 5, which suggests that the Effective Debye Rules are alcohol-dependent. Furthermore, the level of improvement varies between frequencies. Figure 6 illustrates n as a function of volume at 0.2 THz (circles), 0.5 THz (diamonds), and 1.0 THz (squares) for the alcohol solutions, and demonstrates that the largest improvement presents itself in the lower frequencies. The larger deviations at lower THz frequencies correspond to slower rotational dynamics due to disruption of the hydrogen bond network of the pure solvents and the introduction of an inter-species network.¹¹ The improved agreement between the Effective Debye Rules and the measurements further suggests that these Rules can be used to describe the hydrogen-bond networks of binary solutions.

Fig. 6.

A comparison of the refractive indices between the models and the measured values at 0.2 THz (circles), 0.5 THz (diamonds), and 1.0 THz (stars) show the greatest improvement at the lower frequencies for (a) methanol- (b) ethanol-, (c) 1-propanol-, and (d) 2-propanol-water solutions. The deviations between the measured values and the Bruggeman model at lower frequencies corresponds to disturbances in the hydrogen bond network of water.¹¹ The improved agreement between the Effective Debye Rules and the measurements suggest that the predictive effective Debye model is able to capture changes in the hydrogen bond network.

Previous studies have investigated the mixing states,^5–10 per-colation,⁴³ and microheterogeneity^44–47 of various alcohol-water solutions. These are in part governed by self-association dynamics, or the tendency for like molecules to interact with one another within a solution. In aqueous alcohol, this can be affected by steric hindrance due to the chain length of the alkyl group or hydrophobic interactions.^{10,34,48–50}

In comparison to ethanol, 1-propanol, and 2-propanol, methanol is the smallest molecule and more strongly bonds with water.³⁴ These properties allow the hydrogen bond networks of water and methanol to more readily interact. Although the molecular dynamics of methanol-water solutions are well understood, the resulting structure has been a topic of de-bate.^43,51 Some suggest that the stronger hydrogen bonding between methanol and water molecules can create a mixed-species hydrogen bond network.^34,43 In other words, both the chain-like network of methanol, three-dimensional structure of water, and a unique combination cooperatively exist within the solution at some range of volume fractions. Others argue a bi-percolating system does not exist, and that cyclic and linearly branching clusters of methanol molecules are isolated by the hydrogen bond network of water,⁵¹ or that water molecules bridge neighboring methanol molecules.⁵² Regardless, both models agree that methanol molecules can maintain local structures similar to pure methanol, and that water’s dominant hydrogen bond network has an increased lifetime due to hydrophobic effects. This implies that there is an interaction between within-chain methanol molecules with the fast-switching water molecules, yet there are still end-chain molecules that exist within the localized cluster, which agrees with the interpretation of Effective Debye Rule 1.

Structural studies of ethanol-water solution dynamics show that hydrophobic interactions and the complementary relationship in aqueous ethanol cause the ethanol molecules to self-associate.^34,48,53,54 This results in clathrates or “sandwichshaped” ethanol clusters with a hydrophilic surface.⁵³ An individual aggregate moves as a single unit and slides along local water clusters within the solution through the breaking and formation of hydrogen bonds. These water aggregates can also act as bridges between ethanol clusters or other water structures.⁵³ Thus, these water aggregates have within-chain and end-of chain properties that interact with both the edge and inner molecules of stacked ethanol structure. This is in agreement with the derivation of Effective Debye Rule 2, which best fits ethanol solutions.

1-propanol forms zigzag chains,⁵⁰ and it has been suggested that both propyl alcohols share a similar mesoscopic architecture in their pure forms.⁴² It has also been reported that they share similar chain-structures within aqueous solutions.^42,55 Due to the size of the hydrophobic moiety, they more easily disturb water’s tetrahedral network at lower volume fractions than methanol and ethanol, and have a greater tendency to self-associate.^34,42,48,55 This results in both alcohol chains and water clusters to simultaneously exist in aqueous solutions of 1-propanol and 2-propanol. The end chain alcohol monomers can hydrogen bond with the water clusters.

Whether 1-propanol and 2-propanol remain as single chains or create a layered-stacks in solution has not been previously reported. However, Rule 3 of the effective Debye model suggests that the intermediate and fast relaxations of alcohol are indiscernible, and a single τ₂ represents the effective hydrogen bond switching dynamics in water-mixed solution. Therefore, we can assume that water is hydrogen bonding with the within- and edge-chain monomers of the propanol chains equally. This would not be possible with the bridged-and-stacked structure of ethanol clathrates, nor cyclic/linear conformation of methanol clusters.

6. Conclusion

In this paper, an effective media dielectric approximation for binary mixtures of polar liquids has been presented. This model calculates the individual parameters of mixtures, such as the dielectric constants and relaxation times, for use in the Debye model based on the empirical parameters of pure water and several alcohols. The original Debye model was expanded to account for the intermolecular interactions. The theory was validated with THz-TDS reflection measurements of aqueous methanol, ethanol, 1-propanol, and 2-propanol solutions, and shows a significant improvement over traditional effective media models (i.e. Bruggeman Model). Although there is no unifying Effective Debye Rule applicable to all alcohol solutions, it suggests that the Effective Debye Rules reveal the unique nature of the dynamic hydrogen bond network in each solution. The improvements in the predicted dielectric functions suggest that the Effective Debye Rules accurately account for the cooperative motions of the inter-species hydrogen bonds. Comparison between the ideal Effective Debye Rules and the previous studies on alcohol-water interactions further validate the assumptions made in the derivations. Future studies can expand the application of the proposed Effective Debye Rules to other polar liquid or water-solute mixtures. It is anticipated that the length of the hydrocarbon chain would be the more likely predictor of the most suitable Effective Debye Rule. The ideal Effective Debye Rule may also be indicative of the mixture’s hydrogen bond network structure.