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Microfluidics approach for determination of the miscibility gap of multicomponent liquid-liquid systems
Journal Pre-proofs Microfluidics approach for determination of the miscibility gap of multicomponent liquid-liquid systems Martin Hübner, Mirjana Minceva PII: DOI: Reference:
S0894-1777(19)31255-5 https://doi.org/10.1016/j.expthermflusci.2019.109971 ETF 109971
To appear in:
Experimental Thermal and Fluid Science
Received Date: Revised Date: Accepted Date:
2 August 2019 25 September 2019 26 October 2019
Please cite this article as: M. Hübner, M. Minceva, Microfluidics approach for determination of the miscibility gap of multicomponent liquid-liquid systems, Experimental Thermal and Fluid Science (2019), doi: https:// doi.org/10.1016/j.expthermflusci.2019.109971
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Microfluidics approach for determination of the miscibility gap of multicomponent liquid-liquid systems Martin Hübner a and Mirjana Minceva a,* a
Biothermodynamics, TUM School of Life Sciences Weihenstephan, Technical University of Munich, Maximus-von-Imhof-Forum 2, 85354 Freising, Germany
* Corresponding author, E-mail: [email protected]; phone: + 49 8161 71 6170 0. Abstract The principles of microfluidics have been used to develop a novel method to determine compositions on the binodal of a biphasic system. Within this work, a sound theoretical background of the developed method is given, and its experimental applicability is shown. The method is based on a mass balance that correlates the composition of a heterogeneous, multicomponent biphasic system with the position of the phase interface in the microchannel. The proposed method was validated using a system of water, acetone, and toluene. The calculated compositions on the binodal are in a good agreement with published literature. Keywords: binodal composition; parallel, laminar flow; liquid-liquid interface; thermodynamic measurement 1. Introduction Before a novel solvent is able to replace already-used solvents in an existing process, its properties must meet several requirements. In liquid-liquid extraction, information about the phase equilibrium is the most important system property and fundamental prerequisite for any liquidliquid extraction process design [1]. As a preliminary step in process design, fully or semipredictive thermodynamic models [2] may be used as fast screening tools to predict liquid-liquid equilibria in a search for suitable solvents. However, experimental data is always needed to affirm predictions or in cases where thermodynamic models fail to describe the non-ideality of a system [3]. A literature review revealed many methods to determinate liquid-liquid phase equilibria [4-12]. While equilibrium phase compositions of binary liquid-liquid systems may be accessed indirectly or directly by analytical, turbidimetric, and volumetric methods, measuring liquid-liquid equilibrium compositions of ternary or, in particular, multicomponent systems becomes increasingly more
difficult. That means, chemical compositions of (Nc-1)-compounds of each conjugated phase must be determined. Direct analytical methods, such as gas chromatography (GC) or high performance liquid chromatography (HPLC), are used to analyze samples of conjugated phases [13]. Other methods utilize information of the binodal, which separates the homogeneous from the heterogeneous regions, and any other property of this curve (e.g., masses of both conjugated phases [11]) to determine compositions of conjugated phases. Other methods utilize information about the volume and density of conjugated phases to determine the binodal [12]. These methods are often referred to as indirect methods. All given methods share requirements of samples, material, and time for system equilibration as well as sample preparation and analysis. Large sample volumes become considerably more problematic when dealing with toxic solvents or pure compounds that are not readily available. Furthermore, some approaches have been recently proposed that tackle issues of sample volume and time consumption with automation. These methods, however, rely on analyses of samples of conjugated phases by GC that result in a relatively long experimental time [14, 15]. An alternative approach to measure phase equilibria that addresses the need for the reduction of sample size, experimental materials, consumables, and time is the utilization of microfluidic devices [16]. Microfluidics has gained popularity due to its ability of decreasing sample volumes and reaction times under well defined, controlled conditions [17, 18]. A brief literature review reveals the versatility of microfluidic devices. There is a vast number of microfluidic methods and/or devices that measure binodal compositions by either droplet-capturing techniques [19, 20], segmented flows [21, 22], or laminar flows [23, 24]. Within the droplet-capturing techniques, droplets must be generated within a microfluidic set-up and then captured in wells. Subsequently, their properties are analyzed over time using optical, non-invasive methods such as microscopy or fluorescence. The principle behind droplet-capturing techniques is the determination of compositions on the binodal via the use of the commonly-used titration method [19-21]. Instead, of only capturing droplets in wells, it is also possible to dynamically investigate their properties. Therefore, two conjugated phases flowing in a channel in segments are used. The displacement of segments is captured using different non-invasive methods and followed by data analysis [21, 22]. Silva et al. [23] used laminar flow patterns paired with optical observation of the interface to determine binodal compositions of ternary aqueous biphasic systems. Two ternary mixtures were fed on the chip by two syringe pumps that mixed on the chip and separated into two conjugated phases. By setting both volumetric flow rates, the composition of the heterogeneous, multicomponent biphasic system was calculated and related to the optical observations. The group defined the presence of an interface in three different states: no, stable, or unstable interfaces. An unstable interface is characterized by arbitrarily alternating between no interface and different
positions of the formed interface and occurred in the vicinity of the binodal. An empirical function depicting the binodal was then fitted to the data that showed an unstable interface. Recently, a novel method has been proposed by Thien et al. [25] using Raman spectroscopy for the analyses of phase compositions of two parallel, laminar flowing equilibrated phases. Two syringe pumps pumped two partially miscible solutions resulting in two parallel laminar flows. Both phases equilibrated along the channel. The phase composition was determined by Raman spectroscopy at different positions on the chip to ensure that thermodynamic equilibrium had been achieved. Finally, the Raman spectrum of the mixture are used to calculate the concentration of all components in each phase by the indirect hard model (IHM) approach [26, 27]. However, this innovative approach requires additional measurements of known concentrations to fit model parameters that capture the influence of one compound on another. In this work, we have proposed a method to determine the miscibility gap (binodal compositions) for multicomponent systems using a simple microfluidic set-up. A priori, no information on the properties of any compounds nor acquisition of a calibration curve are required in advance of the measurements. The presented system consists of a pulsation-free pressure unit of a pump, microchip, and microscope equipped with a camera. Mixtures are fed on the microchip at different volumetric flow rates. On the chip, the fed mixtures mix and form a parallel, laminar flow. By varying the volumetric flow rates, the position of the phase interface in the microchannel is altered. The position of the phase interface can be determined from images taken with a camera mounted on a microscope. The proposed method utilizes the changes of the location of the phase interface with different compositions of the heterogeneous, multicomponent biphasic system. This data is then correlated with the known compositions of the fed mixtures to calculate compositions on the binodal, eventually. For simplicity, the proposed method is demonstrated for a ternary liquid-liquid system consisting of water, acetone, and toluene. 2. Theoretical background A microfluidic system consists of a series of pulsation-free pumps, a microchip, and a microscope. The flow rates of each pump can be altered at any time. Furthermore, volumetric flows of all pumps and their respective solution properties, such as their compositions, are assumed to be known in advance. All flows are thoroughly mixed in a microchannel on the microchip. A pump can deliver a single (liquid) compound or a homogeneous mixture of the system compounds. Within a certain concentration range, a heterogeneous mixture can be formed and separated into two conjugated phases. Under certain operating conditions, this will result in a parallel laminar flow of two distinct phases. The microchannel is assumed to be sufficiently long enough to guarantee thermodynamic equilibrium between both phases. Due to laminar flow conditions, the mass transfer in the channel
is only governed by diffusion and is perpendicular to the flow direction [28]. The location of the formed interfacial boundary between both conjugated phases can be visually observed by a microscope and analyzed by image processing software at any time and position on the microchip. Furthermore, both phases may be separated at the outlet of the microchip. This enables further analysis of each phase, if desired. Under these assumptions, the described set-up of experiments may be exploited to determine compositions on the binodal which will be elucidated in the upcoming sections. The total mass flow of an 𝑀𝑗 mixture (𝑚𝑀𝑗 ) is a sum of the mass flow rates of the individual pumps: 𝑁𝑝
𝑚 𝑀𝑗 =
∑𝑚
𝑖, 𝑀𝑗
(1)
𝑖
where 𝑚𝑖, 𝑀𝑗 is the mass flow of the 𝑖-th pump and 𝑁𝑝 is the total number of pumps. Altering the flow rates of the individual pumps result in different mixture compositions, leading to a homogeneous or heterogeneous mixture. The subscript 𝑀𝑗 indicates the 𝑗-th mixture. In the case of a heterogeneous mixture, the total mass flow of the mixture 𝑀𝑗 is equal to the sum of the mass flows of the two phases in equilibrium leaving the microchannel: 𝑚𝑀𝑗 = 𝑚𝛼𝑀𝑗 + 𝑚𝛽𝑀𝑗
(2)
where and β represent the two liquid phases in equilibrium, respectively. Equations (1) and (2) may be combined to obtain equation (3): 𝑁𝑝
∑𝑚
𝑖, 𝑀𝑗
= 𝑚𝛼𝑀𝑗 + 𝑚𝛽𝑀𝑗
(3)
𝑖
Since it is often more convenient to measure volumetric flows, the mass flows are converted to volumetric flows: 𝑁𝑝
∑𝜌 𝑉
𝑖 𝑖, 𝑀𝑗
= 𝜌𝛼𝑀𝑗𝑉𝛼𝑀𝑗 + 𝜌𝛽𝑀𝑗𝑉𝛽𝑀𝑗
(4)
𝑖
where 𝜌𝑖 and 𝑉𝑖,𝑀𝑗 represent the density and the volumetric flow of the 𝑖-th pump, respectively, 𝜌𝛼𝑀𝑗 and 𝜌𝛽𝑀𝑗 are the densities and 𝑉𝛼𝑀𝑗 and 𝑉𝛽𝑀𝑗 are the volumetric flows of the and 𝛽 phases leaving the microchannel, respectively. The compound balance for the 𝑘-th compound from the 𝑁𝑐 compound mixture is
𝑁𝑝
𝑤𝑘,𝑀𝑗
(∑ )
𝜌𝑖𝑉𝑖,𝑀𝑗 = 𝑤𝛼𝑘,𝑀𝑗𝜌𝛼𝑀𝑗𝑉𝛼𝑀𝑗 + 𝑤𝛽𝑘,𝑀𝑗𝜌𝛽𝑀𝑗𝑉𝛽𝑀𝑗
(5)
𝑖
where the 𝑤𝑘,𝑀𝑗 represents the overall weight fraction of the 𝑘-th compound in the mixture 𝑀𝑗, and 𝑤𝛼𝑘,𝑀𝑗 and 𝑤𝛽𝑘,𝑀𝑗 are the weight fractions of the 𝑘-th compound in the and 𝛽 conjugated phases, respectively. The overall weight fraction of the 𝑘-th compound in the mixture 𝑀𝑗 can be calculated by using equation (6):
𝑤𝑘,𝑀𝑗 =
∑𝑁𝑝𝑤𝑘,𝑖𝜌𝑖𝑉𝑖,𝑀𝑗 𝑖
(∑𝑁𝑛 𝑤𝑙,𝑛𝜌𝑛𝑉𝑛,𝑀 )
∑ 𝑁𝑐 𝑙
𝑝
(6)
𝑗
where 𝑤𝑘,𝑖 is the weight fraction of the 𝑘-th compound in 𝑖-th flow. Note that this is the weight fraction of a compound per total mass of the mixture 𝑀𝑗. To make use of the location of the interface of the two phases in equilibrium, a flow ratio of the volumetric flow rates of the conjugated phases (𝑓𝑀𝑗) is defined. Consequently, combining equations (4) and (5), and solving for the flow ratio yields the following relationship: 𝑓𝑀𝑗 ≡
𝑉𝛼𝑀𝑗 𝑉𝛽𝑀𝑗
=―
𝜌𝛽𝑀𝑗(𝑤𝛽𝑘,𝑀𝑗 ― 𝑤𝑘,𝑀𝑗) 𝜌𝛼𝑀𝑗(𝑤𝛼𝑘,𝑀𝑗 ― 𝑤𝑘,𝑀𝑗)
(7)
Equation (7) relates the weight fraction of the 𝑘-th compound in each phase and mixture with the phase flow ratio and is only physically meaningful for positive flow ratios. Therefore, either of the two boundary conditions (Equations (8) and (9)) must be fulfilled: for 𝑤𝛼𝑘,𝑀𝑗 ≤ 𝑤𝛽𝑘,𝑀𝑗: 𝑤𝛼𝑘,𝑀𝑗 ≤ 𝑤𝑘,𝑀𝑗 ≤ 𝑤𝛽𝑘,𝑀𝑗
(8)
for 𝑤𝛼𝑘,𝑀𝑗 ≥ 𝑤𝛽𝑘,𝑀𝑗: 𝑤𝛼𝑘,𝑀𝑗 ≥ 𝑤𝑘,𝑀𝑗 ≥ 𝑤𝛽𝑘,𝑀𝑗
(9)
The weight fractions of each phase (𝑤𝛼𝑘,𝑀𝑗 and 𝑤𝛽𝑘,𝑀𝑗) vary with the weight fractions of the mixture 𝑀𝑗 (𝑤𝑘,𝑀𝑗) and, thus, with the changes in flow rates of the pumps. From the definition of the flow ratio 𝑓𝑀𝑗 (equation (7)), it can be seen that the flow ratio goes to zero with increasing volumetric flow rate 𝑉𝛽𝑀𝑗 and decreasing volumetric flow rate 𝑉𝛼𝑀𝑗, or to infinity with increasing volumetric flow rate 𝑉𝛼𝑀𝑗 and decreasing volumetric flow rate 𝑉𝛽𝑀𝑗. This also means that if the weight fractions of the mixture 𝑀𝑗 approach the binodal, or the composition on the binodal, the flow ratio 𝑓𝑀𝑗 approaches zero (𝑤𝑘,𝑀𝑗 = 𝑤𝛽𝑘,𝑀𝑗) or infinity (𝑤𝑘,𝑀𝑗 = 𝑤𝛼𝑘,𝑀𝑗). Using the given definition of the flow ratio, the following two equations can be obtained:
lim 𝑤𝑘,𝑀𝑗 = 𝑤𝛽𝑘,𝑀𝑗
(10)
lim 𝑤𝑘,𝑀𝑗 = 𝑤𝛼𝑘,𝑀𝑗
(11)
𝑓𝑀𝑗→0
𝑓𝑀𝑗→∞
Equations (10) and (11) show that it is possible to exploit information on the volumetric flow rates of the conjugated phases to calculate compositions on the binodal. In Figure 1 (a), a plot of the weight fraction of the 𝑘-th compound at different mixtures 𝑀𝑗 over the logarithmic flow ratio shows a sigmoidal curve. Following the sigmoidal curve toward its limits (zero and infinity), it is possible to obtain the compositions of the 𝑘-th compound on the binodal. Once the compositions on the binodal are known the ternary phase diagram can be drawn (Figure 1 (b)). The ternary biphasic system consisting of water, acetone, and toluene is taken as an example to demonstrate and validate the proposed procedure and is summarized in Figure 1.
(a)
(b)
Figure 1: Demonstration of the measurement and calculation principle. (a) Weight fractions of the 𝑘-th compound at different mixtures 𝑀𝑗 over the flow ratio 𝑓𝑀𝑗. Shown is a ternary biphasic system consisting of water, acetone, and toluene (WAT). Symbols: circles-water, triangles-acetone, and squares-toluene. (b) Ternary phase diagram of WAT system in weight fractions. Triangles indicate the resulting compositions on the binodal. The initial mixtures are represented by circles. As the components mix, the composition is shown as the dashed line. In Figure 1, it is assumed that there are two pumps. The first pump pumps a homogeneous mixture of toluene and acetone (𝑤𝑇𝑜 = 0.38 and 𝑤𝐴𝑐 = 0.62) at a constant flow rate. The second pump is then switched on and pumps a homogeneous mixture of water and acetone (𝑤𝑊 = 0.53 and
𝑤𝐴𝑐 = 0.47). Both mixtures are mixed on the microchip, separated into two conjugated phases along the microchip channel, and thermodynamic equilibrium is attained, assuming the channel is sufficiently long enough. The volumetric flow rate of the second pump is slowly increased in a stepwise manner. The location of the phase interface is recorded after every change of the volumetric flow rate of the second pump. Each flow ratio results in a heterogeneous mixture with a specific overall composition 𝑤𝑘,𝑀𝑗. Furthermore, as shown in Figure 1 (b), both initial mixtures mix along a straight line that connects the initial mixtures. Therefore, when the volumetric flow rate of the second pump is increased, the mixture composition will change along this straight line toward the second initial mixture composition. The composition on the binodal of the water-rich phase can be accessed from the left side of Figure 1 (a). Hence, the flow ratio converges in the binodal composition of the weight fraction of the 𝑘-th compound at zero as proposed in equation (10). Accordingly, the composition of the toluene-rich phase can be accessed on the right side of Figure 1 (a) (see equation (11)). The resulting binodal compositions are 𝑤𝑎𝑞 𝑊 = 0.503 and 𝑜𝑟𝑔 𝑜𝑟𝑔 𝑤𝑎𝑞 𝐴𝑐 = 0.478 for the aqueous phase, and 𝑤𝑊 = 0.059 and 𝑤𝐴𝑐 = 0.600 organic phase,
respectively. This procedure must be repeated with different initial mixtures to determine other compositions on the binodal and, thus, to determine the complete binodal. 2.1.
Approximation of the flow ratio
In the previous section, we developed a method that utilizes information on the ratio of volumetric flow rates of the conjugated phases (𝑓𝑀𝑗) in a microchannel to calculate compositions on the binodal for a biphasic system. Within this method, it was assumed that the flow rates of the conjugated phases of a mixture 𝑀𝑗 can be measured and, hence, the composition of a heterogeneous, multicomponent biphasic system can be calculated from the mass balance (equation (6)). A non-invasive measurement of the flow rates in equilibrium, however, is not trivial, even though there are several different methods described in literature [29-35]. Many flow measurements are droplet-based and, consequently, inappropriate for the proposed method, which assumes two translucent, laminar, parallel flows separated by an interface. Therefore, in this work, the flow ratio will be estimated as follows. In Figure 2, a microchannel with a biphasic flow of two equilibrated phases and the phase interface are schematically presented.
(a)
(b)
Figure 2: Schematic presentation of a microfluidic channel. The phase interface between two equilibrated liquid phases is given by a black dashed line. The red dashed line is meant to guide the readers eye. (a) The microchannel is observed as seen by the microscope. The flow pattern is parallel and laminar. (b) Cross section of a microchannel. Considering Figure 2, the flow ratio 𝑓𝑀𝑗 defined in equation (7) can be written in terms of the velocities of the phases: 𝑓𝑀𝑗 ≡
𝑉𝛼𝑀𝑗 𝑉𝛽𝑀𝑗
=
𝑣𝛼𝑀𝑗𝐴𝛼𝑀𝑗 𝑣𝛽𝑀𝑗𝐴𝛽𝑀𝑗
=
𝑣𝛼𝑀𝑗𝑙𝛼𝑀𝑗 𝑣𝛽𝑀𝑗𝑙𝛽𝑀𝑗
(12)
where 𝑣𝑀𝑗, 𝐴𝑀𝑗, and 𝑙𝑀𝑗 are the linear velocity, flow cross sectional area, and distance between the interface and the wall of the microchannel for each phase of a mixture 𝑀𝑗, respectively. For moderate differences in density and viscosity, it has been shown that the velocity profile of both phases can be approximated by a velocity profile that one hypothetical fluid would experience when flowing alone at the same flowing conditions [36]. Therefore, the velocity of both phases is assumed to be equal. With this assumption, the flow ratio 𝑓𝑀𝑗 can be approximated by the ratio of the distances between the phase interface and the microchannel walls, which can be determined by visual observation using a microscope. The flow ratio is then given by: 𝑓𝑀𝑗 ≈
𝑙𝛼𝑀𝑗 𝑙𝛽𝑀𝑗
(13)
Equation (13) provides an easier determination of the compositions on the binodal using equation (7), since the phase flow ratio can be easily calculated utilizing an image processing software to determine the distance between the interface and the wall of the microchannel for each phase. 2.2.
Calculation of binodal composition
In Figure 1 (a), the overall composition of the heterogeneous mixture is shown as a function of the flow ratio 𝑓𝑀𝑗. The weight fraction of the 𝑘-th compound in mixture 𝑀𝑗 is a sigmoidal shape similar to the function of the flow ratio that is bound to conditions given by equations (10) and (11). In literature, sigmoidal functions are often used to model bacteria growth [37]. Within this work, a sigmoidal-like function with four coefficients for each compound has been defined and is given in equation (14): 𝑤𝑘,𝑀𝑗 (𝑓𝑀𝑗) =
𝐴𝑘 1 + exp (𝐵𝑘ln (𝑓𝑀𝑗) ― 𝐶𝑘)
+ 𝐷𝑘
(14)
where 𝐴𝑘, 𝐵𝑘, 𝐶𝑘, and 𝐷𝑘 are adjustable parameters for each compound in mixture 𝑀𝑗. The four coefficients of the function are fitted to the measured data to determine compositions on the binodal of the 𝑘-th compound by extrapolating to zero or infinity. To reduce the computational complexity, coefficients of equation (14) are fitted to the experimental data of (𝑁𝑐 ―1) compounds. The weight fraction of the 𝑁𝑐-th compound can be calculated as follows: 𝑁𝑐 ― 1
𝑤𝑚,𝑀𝑗 = 1 ―
∑𝑤
𝑘,𝑀𝑗
(15)
𝑘≠𝑚
The weight fraction of the 𝑁𝑐-th compound in mixture 𝑀𝑗 can be chosen arbitrarily. In the following, the weight fraction of the 𝑁𝑐-th compound in mixture 𝑀𝑗 is acetone. Equation (14) can be extended to account for more asymmetrical trends. This, however, implies using a sophisticated numerical fitting procedure of all adjustable parameters. In addition, adjustable parameters must be sufficiently constrained so that calculated results are physically meaningful. For simplicity, equation (14) has been applied throughout this work. Regarding equation (14), the following physical meaningful constraints can be derived for 𝐴𝑘,𝐵𝑘 > 0. lim 𝑤𝑘,𝑀𝑗 = 𝐴𝑘 + 𝐷𝑘 = 𝑤𝛽𝑘,𝑀𝑗 min (𝑤𝑘,𝑖) ≤ 𝑤𝛽𝑘,𝑀𝑗 ≤ max (𝑤𝑘,𝑖)
(16)
lim 𝑤𝑘,𝑀𝑗 = 𝐷𝑘 = 𝑤𝑘,𝑀𝑗 min (𝑤𝑘,𝑖) ≤ 𝑤𝑘,𝑀𝑗 ≤ max (𝑤𝑘,𝑖)
(17)
𝑓𝑀𝑗→0
𝑓𝑀𝑗→∞
where 𝑤𝑘,𝑖 represents the weight fraction of the 𝑘-th compound in the flow delivered by the 𝑖-th pump. It should be noted that fed flows on the microchip by any pump may be pure solvent or any homogeneous mixture of the solvent system. The minimum and maximum weight fractions of the 𝑘-th compound determine the compositional boundaries. The calculated binodal weight fraction of each compound must obey the constraints given in equations (16) and (17). 3. Experimental section
3.1.
Materials
Acetone (CAS# 67-64-1) was purchase from VWR International GmbH (Darmstadt, Germany) with a purity of at least 99%. Toluene (CAS# 108-88-3) was supplied by Merck KGaA (Darmstadt, Germany) and has a purity of >99.9%. Milli-Q water (18.2 MΩ cm at 25 °C) was used in all conducted experiments. Isopropanol (CAS# 67-63-0) obtained from VWR International GmbH (Darmstadt, Germany) was used to clean the set-up after each experiment. 3.2.
Microfluidic set-up
The microfluidic set-up is schematically shown in Figure 3. The pure solvents, or homogeneous solvent mixtures were placed in the 100 mL reservoirs. Oil-free, pressurized air was used to apply pressure above the surface of a liquid inside the reservoirs by a flow controller (OB1 Base Mk3+, Elveflow, France). Due to this pressure, liquid was pushed through a tube that had been submerged into the liquid toward a flow rate sensor (MINI CORI-FLOW BFS1, Bronkhorst, Netherland). The flow rate sensor was connected to the microchip inlets. The Y-Y microchip made of glass was manufactured by Micronit Microtechnologies B.V. (Netherland). Its channel length, width, and depth were 3 cm, 400 µm, and 50 µm, respectively. The volumetric flow rate was controlled by a feedback loop of the flow rate sensor to the flow controller. The set-up enabled a precise, pulsation-free, and stable volumetric flow rate. The outlets of the microchip were connected to a waste container. The temperature was checked using a thermometer (digital Minithermometer, VWR Collection, Germany) at the outlet. An inverse microscope (Wilovert standard ph 20, Helmut Hund GmbH, Germany) was used to track the phase interface.
Figure 3: Schematic representation of the microfluidic set-up. 3.3.
Experimental procedure
In this work, only pre-equilibrated phases were pumped on the microchip. Therefore, a mixture composition within the biphasic region was selected and prepared in a sealable container. The mixture was thoroughly mixed using a magnetic stirrer for approximately two hours. After that, the mixture was transferred to a separating funnel where it equilibrated for approximately two hours. The room temperature was maintained at 20 °C. Both phases were separated and filled in reservoirs. Compositions of both phases were determined by gas chromatography (GC) using a thermal conductivity detector after each experiment. The GC method is described in section 3.4. The channel inlets 1 and 2 were fed with the water-rich and toluene-rich phases, respectively. The volumetric flow rate of the water-rich phase was held at a constant value. After the microchip was filled with the water-rich phase, the toluene-rich phase was fed to the microchip at a lower initial volumetric flow rate resulting in a small volumetric flow rate ratio which defined the starting point for all experiments. The flow rate of the toluene-rich phase was increased stepwise, which resulted in different locations of the interface. After each stepwise increase of the toluene-rich phase, the volumetric flow rates were held constant for at least two minutes to ensure that the location of the interface did not change further. When the location of the interface remained constant, an image was taken and the volumetric flow rate of the toluene-rich phase was increased again. It should be mentioned that a stable laminar, parallel flow pattern occurs in a certain range of the volumetric
flow rate ratio, and depends on the system properties. Outside of this range, droplet or slug flow patterns were observed. Therefore, as soon as no stable parallel flow was observed, the initially– selected water-rich phase flow rate was increased. The maximum applied volumetric flow rate of the water- and toluene-rich phases were 800 and 1000 µL min−1, respectively. Furthermore, the minimum and maximum ratios of the volumetric flow rates were 0.02 and 32, respectively. Note that these are ratios determined from set volumetric flow rates of the toluene-rich to water-rich phase. 3.4.
GC analysis
A GC from Shimadzu (Nexis GC 2030) with a thermal conductivity detector (TCD) in split mode was used to analyze the phase composition of the samples taken at the end of each experiment from each reservoir. THF was used to dilute the samples. A Capillary column (30 m length, 0.25 mm ID, and 1.4 µm film thickness) obtained from Restek (Rxi®-624Sil MS) was used for the analysis. Helium was used as the carrier gas with a linear velocity of 40 cm s−1. The temperature of the injection port was set to 250 °C at a split-ratio of 50. To separate water and toluene, an oven temperature gradient was applied after an isothermal step at one minute at 35 °C. The oven temperature gradient was linear and increased 31 °C min−1 to 190 °C. The TCD temperature was set to 260 °C. Samples of both equilibrated phases were placed in the two reservoirs (see Figure 3) and taken at the end of each experiment to analyze their compositions by GC. The determined phase compositions were used to calculate the weight fractions of the 𝑘-th compounds for mixtures 𝑀𝑗, according to Equation (6). Note that analyses by GC are unnecessary for the proposed method since pure solvents or mixtures with known compositions are intended to be used as mixtures fed on the microchip and are not the same as the flow ratios approximated by Equation (13). 3.5.
Determination of the flow ratio
The location of the interfacial boundary was determined by using images obtained with a 5 MP camera with a 1/2'' CMOS-sensor (dhs GmbH, Germany) that was mounted on the inverse microscope. The analyses were done post-experimentally by using image processing toolboxes from MATLAB (version R2017b, MathWorks Inc., USA). The locations of the channel boundaries and interface were then used to calculate the flow ratio (equation (13)). In Figure 4, a typical image taken by the camera mounted on the microscope is shown.
(a)
(b)
(c)
Figure 4: Typical image of the micro channel inlet, middle, and outlet before image treatment. As expected, since the two phases are pre-equilibrated before being fed on the microchip, the position of the phase interphase is constant along the channel of the microchip. Images taken by a camera mounted on the microscope were treated so that the flow ratio could be calculated from the ratio of distances of the phase interface to the channel wall (see section 2.1). Before the images were converted into complementary, binary images, all images were first transformed to gray scale and a Gaussian filter was applied. After that, all blobs were removed and a morphological operation of the microchip and phase interphase to a skeleton structure was performed. The developed image treatment allows a direct determination of the phase interface and the channel walls. Due to perturbations at the outlet of the channel, the last location at which the phase interface was determined was at about 60% of the total channel length. For experiments with non-pre-equilibrated phases, the following should be considered when selecting the position for taking images for the determination of the interface position: (i) the system should have reached equilibrium and (ii) the position of the phase interface should not be influenced by the channel outlet. All images were treated according to the proposed image treatment procedure. The flow ratio was then determined by the ratio of the determined lengths of the toluene-rich to the water-rich phase (see Equation (13)). 4. Results and discussion In this work, a novel method based on mass balances was proposed to determine compositions on the binodal utilizing a microfluidic set-up (see Figure 3). The theoretical background of the method was described in section 2. For experimental validation, a WAT system was selected. This system is used as a standard system in liquid-liquid extraction research, and its liquid-liquid equilibrium, including the density, dynamic viscosity, and interfacial tension, has been already extensively measured and documented. Correlations for the calculation of the binodal and phase properties are available in literature [38]. Additionally, to avoid unnecessary discussions about whether or not the length of the microchannel is sufficiently long enough for the biphasic system to reach thermodynamic
equilibrium, pre-equilibrated phases were used to validate the proposed procedure. As already mentioned above, mass transfer is only governed by diffusion in the perpendicular direction of flow. Hence, if both phases are already in thermodynamic equilibrium, mass transfer can be neglected. This experimental procedure has been chosen to focus on the determination of the binodal compositions. Nevertheless, the proposed method is not limited to the use of preequilibrated phases. In case of non pre-equilibrated phases, the length of the microchannel needs to be sufficiently long to guarantee thermodynamic equilibrium between both phases at the outlet of the microchip. A minimum length of the microchannel can be roughly estimated by dividing the product of the mean phase velocity and the squared channel width by the diffusion coefficient of acetone in water or toluene. Four systems were prepared volumetrically as described in section 3.3. Their compositions are given in Table 1. Table 1: Volumetric compositions of conducted experiments. Interfacial tensions were calculated from a correlation from [38].
System
WAT (v/v/v)
[mN m−1]
I
47.5/5/47.5
26.4
II
40/20/40
12.4
III
30/40/30
4.8
IV
20/60/20
0.9
For all mixtures, the ratio of water to toluene was held constant (1:1). Only the acetone content increased from system I to IV. In preliminary experiments, it was found that the standard error of between repeated measurements was sufficiently small. Hence, all experiments were conducted once. Before any experiment was conducted, equilibrated phases were prepared according to the described procedure in section 3.3. Both phases were then fed to the chip. The flow rate of the water-rich phase was kept constant, while the toluene-rich phase was increased stepwise. Under these conditions, a laminar, parallel flow pattern was observed. A biphasic flow and, consequently, the phase interface in a microchannel, was determined by interfacial tension and inertial and viscous forces. To characterize the flow, dimensionless numbers were used, namely Reynolds (inertial/frictional forces), Weber (inertial forces/interfacial tension), and Capillary numbers (frictional forces/interfacial tension). The ranges of the Reynolds, Weber, and Capillary numbers are summarized in Table 2.
Table 2: Ranges of dimensionless numbers, Reynolds, Capillary, and Weber numbers, for all conducted experiments. Compositions of systems I to IV are given in Table 1. No.
Reaq
Reorg
I
9.7–87
II
1.2–8.6
4.5–64.4
III
1.0–12.9
IV
0.9–6.1
Caaq
Caorg
Weaq
Weorg
3.9 10−3–2.9 10−2
0.05–3.8
0.08–3.9
2.0 10−3–1.5 10−2
1.5 10−3–1.9 10−2
0.00–0.1
0.01–1.2
4.8–64.0
5.6 10−3–7.5 10−2
4.9 10−3–5.7 10−2
0.01–1.0
0.02–3.7
3.7–51.8
3.2 10−2–2.2 10−1
3.0 10−2–3.7 10−1
0.03–1.3
0.11–19
19.6–134.5 5.0 10−3–4.4 10−2
The calculated Reynolds numbers indicate a laminar flow, and the Capillary and Weber numbers indicate a parallel flow in the microchannel, which is in accordance to observations for a microchannel with a rectangular cross section [39, 40]. It can be noted that the interfacial tension decreases exponentially with increasing acetone content toward the plait point (𝑤𝑊 = 0.254, 𝑤𝐴𝑐 = 0.637, and 𝑤𝑇 = 0.109 [38]), from systems I to IV (see Table 1). In attaining the plait point, the phase interfacial tension vanishes since the biphasic system collapses to one phase. Due to this fact the proposed method is not capable of measuring the plait point composition. The weight fractions of the 𝑘-th compounds for mixtures 𝑀𝑗 versus the determined flow ratios 𝑓𝑀𝑗 from equation (13), for the four studied systems, are shown in Figure 5.
(a) 47.5/5/47.5
(b) 40/20/40
(c) 30/40/30
(d) 20/60/20
Figure 5: Weight fractions of water and toluene as a function of the flow ratio for systems I-IV listed in Table 1. In Figure 5, the flow ratios 𝑓𝑀𝑗 for each mixture 𝑀𝑗 were determined from images using the approximation given in equation (13) (symbols). The weight fractions of water and toluene as a function of flow ratios were fitted with equation (14). The coefficients of equation (14) were fitted to experimental data following the conditions given in equations (16) and (17). Dashed lines in Figure 5 represent a 95% confidence interval of the fitted coefficients. As demonstrated in Figure 5, the fitted functions are in a good agreement (𝑅2 > 0.99) with the experimental data. This means that equation (14) is reasonable for a phenomenological description of the experimental data. The calculated compositions on the binodal are shown in Figure 6 for the four initial compositions given in Table 1.
Figure 6: Phase diagram of water, acetone, and toluene. The solid line represents binodal compositions calculated from a correlation taken from literature [38]. The circles represent calculated compositions on the binodal by the proposed method and are connected through dashed–dotted lines to indicate the simultaneous calculations of binodal compositions in the water- and toluene-rich phases. (AADW = 0.0017, AADT = 0.0013, AAD = 0.0015) To evaluate the precision of the proposed method, the minimal distance 𝑑𝑚𝑖𝑛 between the experimentally-determined phase composition with the proposed method and the calculated phase composition using the experimentally obtained correlation from Mísek et al. [38] was defined as 𝑁𝐶
𝑑𝑚𝑖𝑛 =
∑(𝑤
𝑒𝑥𝑝 𝑘
2
― 𝑤𝑙𝑖𝑡 𝑘 )
(18)
𝑘
where 𝑤𝑒𝑥𝑝 is the experimentally-determined weight fraction of the 𝑘-th compound. 𝑤𝑙𝑖𝑡 𝑘 𝑘 is the weight fraction of the 𝑘-th compound calculated using the correlation taken from literature [38]. The value of 𝑤𝑙𝑖𝑡 𝑘 was iteratively calculated to minimize equation (18). An absolute average deviation (AAD) between both weight fractions was calculated using equation (19): 𝑚
AAD =
𝑝
𝑐
∑ ∑ ∑ |𝑤
𝑙𝑖𝑡,𝑗 𝑘,𝑙
𝑙 = 1𝑗 = 1𝑘 = 1
― 𝑤𝑒𝑥𝑝,𝑗 𝑘,𝑙 |
(19)
where 𝑐, 𝑝, and 𝑚 represent the number of compounds in the system, phases, and measured 𝑒𝑥𝑝,𝑗 points, respectively. 𝑤𝑙𝑖𝑡,𝑗 𝑘,𝑙 and 𝑤𝑘,𝑙 are literature and experimentally-determined weight fractions
of the 𝑘-th compounds in the 𝑗-th phases of the 𝑙-th data, respectively. Compositions on the binodal in both phases, water- and toluene-rich phases, are in an overall good agreement with literature (AADW = 0.0017, AADT = 0.0013, AAD = 0.0015). Finally, the experimental data were fitted by the following empirical equation that is often used in literature to fit cloud-point titration data of aqueous biphasic systems [11]. 3 𝑤𝑇 = 𝐴exp {𝐵𝑤0.5 𝑊 + 𝐶𝑤𝑊}
(20)
𝐴, 𝐵, and 𝐶 are adjustable parameters that are fitted to the determined binodal compositions. In Figure 7, the fitted binodal is in a good agreement with our experimental data (𝑅2 > 0.99) and data taken from literature [38].
Figure 7: Phase diagram of water, acetone, and toluene. The solid line represents binodal compositions calculated from a correlation taken from literature [38]. The dashed line represents results for the fit of equation (20) to determine binodal compositions. It is shown that the binodal can be accurately described with a few precise measurements. Hence, the proposed method can be used for the determination of the binodal and miscibility gap of multicomponent liquid-liquid systems. 5. Conclusion
In this work, a novel method to determine compositions on the binodal of multicomponent mixtures was theoretically developed and experimentally validated for a ternary system. The proposed method utilized a microfluidic set-up with a pulsation-free pump paired with flow meters controlled by a feedback loop to the pressure unit. This allows the microchip to be fed with a stable and precise volumetric flow rate so that the overall mass on the chip of any compound is known and, furthermore, able to be defined. The set-up reduces the solvent and time consumption. Additionally, it eases handling toxic compounds due to a closed system. The proposed method is based on the assumption of a stable laminar, parallel flow pattern. Therefore, the volumetric flow rates at the inlets of the microchip should be selected to ensure that a stable laminar, parallel flow pattern is obtained in the microchannel. A simple extrapolation procedure has been developed to determine the compositions on the binodal by using the information on the volumetric flow rates and the position of the phase interface of the two equilibrated phases in the microchannel. The overall composition of the system is determined by setting the flow rates of inlet streams consisting of pure component or a homogeneous mixture of the system components that result in changes of the position of the phase interface. An image processing procedure was developed to determine the position of the phase interface in the microchannel by a suitable approximation of the resulting flow ratio. Compositions on the binodal were governed in two steps. First, a sigmoidal function was fitted to overall weight fractions of the 𝑘-th compound at mixture 𝑀𝑗 as a function of approximated flow ratios. After that, the sigmoidal functions were extrapolated to zero and infinity, respectively, to determine the compositions in each respective phase. As an extension of the proposed method, information on the binodal compositions may be used in a subsequent step to calculate the phase equilibrium compositions by measuring an additional property of one of the phases at each set flow rate. Therefore, this method is a promising for the simultaneous determination of the binodal and tie lines. The composition of a liquid-liquid system in equilibrium could be characterized with a few experiments using less solvent than in conventional methods. 6. Acknowledgment This work is part of a cooperation project that is funded by the Bayrische Forschungsstiftung (AZ1285-17) to promote and strengthen the Bavarian position in science and technology. This work was supported by the German Research Foundation (DFG) and the Technical University of Munich within the funding program Open Access Publishing. The authors are thankful the financial support.
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Conflict of interest The authors declare no conflict of interest.
Highlights
A novel microfluidic approach for determination of binodal compositions is proposed
A laminar, parallel flow is observed in the microchannel
Position of the phase interface is captured by an inverse microscope
Determined miscibility gab is in a good agreement with literature data
S0894-1777(19)31255-5 https://doi.org/10.1016/j.expthermflusci.2019.109971 ETF 109971
To appear in:
Experimental Thermal and Fluid Science
Received Date: Revised Date: Accepted Date:
2 August 2019 25 September 2019 26 October 2019
Please cite this article as: M. Hübner, M. Minceva, Microfluidics approach for determination of the miscibility gap of multicomponent liquid-liquid systems, Experimental Thermal and Fluid Science (2019), doi: https:// doi.org/10.1016/j.expthermflusci.2019.109971
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Microfluidics approach for determination of the miscibility gap of multicomponent liquid-liquid systems Martin Hübner a and Mirjana Minceva a,* a
Biothermodynamics, TUM School of Life Sciences Weihenstephan, Technical University of Munich, Maximus-von-Imhof-Forum 2, 85354 Freising, Germany
* Corresponding author, E-mail: [email protected]; phone: + 49 8161 71 6170 0. Abstract The principles of microfluidics have been used to develop a novel method to determine compositions on the binodal of a biphasic system. Within this work, a sound theoretical background of the developed method is given, and its experimental applicability is shown. The method is based on a mass balance that correlates the composition of a heterogeneous, multicomponent biphasic system with the position of the phase interface in the microchannel. The proposed method was validated using a system of water, acetone, and toluene. The calculated compositions on the binodal are in a good agreement with published literature. Keywords: binodal composition; parallel, laminar flow; liquid-liquid interface; thermodynamic measurement 1. Introduction Before a novel solvent is able to replace already-used solvents in an existing process, its properties must meet several requirements. In liquid-liquid extraction, information about the phase equilibrium is the most important system property and fundamental prerequisite for any liquidliquid extraction process design [1]. As a preliminary step in process design, fully or semipredictive thermodynamic models [2] may be used as fast screening tools to predict liquid-liquid equilibria in a search for suitable solvents. However, experimental data is always needed to affirm predictions or in cases where thermodynamic models fail to describe the non-ideality of a system [3]. A literature review revealed many methods to determinate liquid-liquid phase equilibria [4-12]. While equilibrium phase compositions of binary liquid-liquid systems may be accessed indirectly or directly by analytical, turbidimetric, and volumetric methods, measuring liquid-liquid equilibrium compositions of ternary or, in particular, multicomponent systems becomes increasingly more
difficult. That means, chemical compositions of (Nc-1)-compounds of each conjugated phase must be determined. Direct analytical methods, such as gas chromatography (GC) or high performance liquid chromatography (HPLC), are used to analyze samples of conjugated phases [13]. Other methods utilize information of the binodal, which separates the homogeneous from the heterogeneous regions, and any other property of this curve (e.g., masses of both conjugated phases [11]) to determine compositions of conjugated phases. Other methods utilize information about the volume and density of conjugated phases to determine the binodal [12]. These methods are often referred to as indirect methods. All given methods share requirements of samples, material, and time for system equilibration as well as sample preparation and analysis. Large sample volumes become considerably more problematic when dealing with toxic solvents or pure compounds that are not readily available. Furthermore, some approaches have been recently proposed that tackle issues of sample volume and time consumption with automation. These methods, however, rely on analyses of samples of conjugated phases by GC that result in a relatively long experimental time [14, 15]. An alternative approach to measure phase equilibria that addresses the need for the reduction of sample size, experimental materials, consumables, and time is the utilization of microfluidic devices [16]. Microfluidics has gained popularity due to its ability of decreasing sample volumes and reaction times under well defined, controlled conditions [17, 18]. A brief literature review reveals the versatility of microfluidic devices. There is a vast number of microfluidic methods and/or devices that measure binodal compositions by either droplet-capturing techniques [19, 20], segmented flows [21, 22], or laminar flows [23, 24]. Within the droplet-capturing techniques, droplets must be generated within a microfluidic set-up and then captured in wells. Subsequently, their properties are analyzed over time using optical, non-invasive methods such as microscopy or fluorescence. The principle behind droplet-capturing techniques is the determination of compositions on the binodal via the use of the commonly-used titration method [19-21]. Instead, of only capturing droplets in wells, it is also possible to dynamically investigate their properties. Therefore, two conjugated phases flowing in a channel in segments are used. The displacement of segments is captured using different non-invasive methods and followed by data analysis [21, 22]. Silva et al. [23] used laminar flow patterns paired with optical observation of the interface to determine binodal compositions of ternary aqueous biphasic systems. Two ternary mixtures were fed on the chip by two syringe pumps that mixed on the chip and separated into two conjugated phases. By setting both volumetric flow rates, the composition of the heterogeneous, multicomponent biphasic system was calculated and related to the optical observations. The group defined the presence of an interface in three different states: no, stable, or unstable interfaces. An unstable interface is characterized by arbitrarily alternating between no interface and different
positions of the formed interface and occurred in the vicinity of the binodal. An empirical function depicting the binodal was then fitted to the data that showed an unstable interface. Recently, a novel method has been proposed by Thien et al. [25] using Raman spectroscopy for the analyses of phase compositions of two parallel, laminar flowing equilibrated phases. Two syringe pumps pumped two partially miscible solutions resulting in two parallel laminar flows. Both phases equilibrated along the channel. The phase composition was determined by Raman spectroscopy at different positions on the chip to ensure that thermodynamic equilibrium had been achieved. Finally, the Raman spectrum of the mixture are used to calculate the concentration of all components in each phase by the indirect hard model (IHM) approach [26, 27]. However, this innovative approach requires additional measurements of known concentrations to fit model parameters that capture the influence of one compound on another. In this work, we have proposed a method to determine the miscibility gap (binodal compositions) for multicomponent systems using a simple microfluidic set-up. A priori, no information on the properties of any compounds nor acquisition of a calibration curve are required in advance of the measurements. The presented system consists of a pulsation-free pressure unit of a pump, microchip, and microscope equipped with a camera. Mixtures are fed on the microchip at different volumetric flow rates. On the chip, the fed mixtures mix and form a parallel, laminar flow. By varying the volumetric flow rates, the position of the phase interface in the microchannel is altered. The position of the phase interface can be determined from images taken with a camera mounted on a microscope. The proposed method utilizes the changes of the location of the phase interface with different compositions of the heterogeneous, multicomponent biphasic system. This data is then correlated with the known compositions of the fed mixtures to calculate compositions on the binodal, eventually. For simplicity, the proposed method is demonstrated for a ternary liquid-liquid system consisting of water, acetone, and toluene. 2. Theoretical background A microfluidic system consists of a series of pulsation-free pumps, a microchip, and a microscope. The flow rates of each pump can be altered at any time. Furthermore, volumetric flows of all pumps and their respective solution properties, such as their compositions, are assumed to be known in advance. All flows are thoroughly mixed in a microchannel on the microchip. A pump can deliver a single (liquid) compound or a homogeneous mixture of the system compounds. Within a certain concentration range, a heterogeneous mixture can be formed and separated into two conjugated phases. Under certain operating conditions, this will result in a parallel laminar flow of two distinct phases. The microchannel is assumed to be sufficiently long enough to guarantee thermodynamic equilibrium between both phases. Due to laminar flow conditions, the mass transfer in the channel
is only governed by diffusion and is perpendicular to the flow direction [28]. The location of the formed interfacial boundary between both conjugated phases can be visually observed by a microscope and analyzed by image processing software at any time and position on the microchip. Furthermore, both phases may be separated at the outlet of the microchip. This enables further analysis of each phase, if desired. Under these assumptions, the described set-up of experiments may be exploited to determine compositions on the binodal which will be elucidated in the upcoming sections. The total mass flow of an 𝑀𝑗 mixture (𝑚𝑀𝑗 ) is a sum of the mass flow rates of the individual pumps: 𝑁𝑝
𝑚 𝑀𝑗 =
∑𝑚
𝑖, 𝑀𝑗
(1)
𝑖
where 𝑚𝑖, 𝑀𝑗 is the mass flow of the 𝑖-th pump and 𝑁𝑝 is the total number of pumps. Altering the flow rates of the individual pumps result in different mixture compositions, leading to a homogeneous or heterogeneous mixture. The subscript 𝑀𝑗 indicates the 𝑗-th mixture. In the case of a heterogeneous mixture, the total mass flow of the mixture 𝑀𝑗 is equal to the sum of the mass flows of the two phases in equilibrium leaving the microchannel: 𝑚𝑀𝑗 = 𝑚𝛼𝑀𝑗 + 𝑚𝛽𝑀𝑗
(2)
where and β represent the two liquid phases in equilibrium, respectively. Equations (1) and (2) may be combined to obtain equation (3): 𝑁𝑝
∑𝑚
𝑖, 𝑀𝑗
= 𝑚𝛼𝑀𝑗 + 𝑚𝛽𝑀𝑗
(3)
𝑖
Since it is often more convenient to measure volumetric flows, the mass flows are converted to volumetric flows: 𝑁𝑝
∑𝜌 𝑉
𝑖 𝑖, 𝑀𝑗
= 𝜌𝛼𝑀𝑗𝑉𝛼𝑀𝑗 + 𝜌𝛽𝑀𝑗𝑉𝛽𝑀𝑗
(4)
𝑖
where 𝜌𝑖 and 𝑉𝑖,𝑀𝑗 represent the density and the volumetric flow of the 𝑖-th pump, respectively, 𝜌𝛼𝑀𝑗 and 𝜌𝛽𝑀𝑗 are the densities and 𝑉𝛼𝑀𝑗 and 𝑉𝛽𝑀𝑗 are the volumetric flows of the and 𝛽 phases leaving the microchannel, respectively. The compound balance for the 𝑘-th compound from the 𝑁𝑐 compound mixture is
𝑁𝑝
𝑤𝑘,𝑀𝑗
(∑ )
𝜌𝑖𝑉𝑖,𝑀𝑗 = 𝑤𝛼𝑘,𝑀𝑗𝜌𝛼𝑀𝑗𝑉𝛼𝑀𝑗 + 𝑤𝛽𝑘,𝑀𝑗𝜌𝛽𝑀𝑗𝑉𝛽𝑀𝑗
(5)
𝑖
where the 𝑤𝑘,𝑀𝑗 represents the overall weight fraction of the 𝑘-th compound in the mixture 𝑀𝑗, and 𝑤𝛼𝑘,𝑀𝑗 and 𝑤𝛽𝑘,𝑀𝑗 are the weight fractions of the 𝑘-th compound in the and 𝛽 conjugated phases, respectively. The overall weight fraction of the 𝑘-th compound in the mixture 𝑀𝑗 can be calculated by using equation (6):
𝑤𝑘,𝑀𝑗 =
∑𝑁𝑝𝑤𝑘,𝑖𝜌𝑖𝑉𝑖,𝑀𝑗 𝑖
(∑𝑁𝑛 𝑤𝑙,𝑛𝜌𝑛𝑉𝑛,𝑀 )
∑ 𝑁𝑐 𝑙
𝑝
(6)
𝑗
where 𝑤𝑘,𝑖 is the weight fraction of the 𝑘-th compound in 𝑖-th flow. Note that this is the weight fraction of a compound per total mass of the mixture 𝑀𝑗. To make use of the location of the interface of the two phases in equilibrium, a flow ratio of the volumetric flow rates of the conjugated phases (𝑓𝑀𝑗) is defined. Consequently, combining equations (4) and (5), and solving for the flow ratio yields the following relationship: 𝑓𝑀𝑗 ≡
𝑉𝛼𝑀𝑗 𝑉𝛽𝑀𝑗
=―
𝜌𝛽𝑀𝑗(𝑤𝛽𝑘,𝑀𝑗 ― 𝑤𝑘,𝑀𝑗) 𝜌𝛼𝑀𝑗(𝑤𝛼𝑘,𝑀𝑗 ― 𝑤𝑘,𝑀𝑗)
(7)
Equation (7) relates the weight fraction of the 𝑘-th compound in each phase and mixture with the phase flow ratio and is only physically meaningful for positive flow ratios. Therefore, either of the two boundary conditions (Equations (8) and (9)) must be fulfilled: for 𝑤𝛼𝑘,𝑀𝑗 ≤ 𝑤𝛽𝑘,𝑀𝑗: 𝑤𝛼𝑘,𝑀𝑗 ≤ 𝑤𝑘,𝑀𝑗 ≤ 𝑤𝛽𝑘,𝑀𝑗
(8)
for 𝑤𝛼𝑘,𝑀𝑗 ≥ 𝑤𝛽𝑘,𝑀𝑗: 𝑤𝛼𝑘,𝑀𝑗 ≥ 𝑤𝑘,𝑀𝑗 ≥ 𝑤𝛽𝑘,𝑀𝑗
(9)
The weight fractions of each phase (𝑤𝛼𝑘,𝑀𝑗 and 𝑤𝛽𝑘,𝑀𝑗) vary with the weight fractions of the mixture 𝑀𝑗 (𝑤𝑘,𝑀𝑗) and, thus, with the changes in flow rates of the pumps. From the definition of the flow ratio 𝑓𝑀𝑗 (equation (7)), it can be seen that the flow ratio goes to zero with increasing volumetric flow rate 𝑉𝛽𝑀𝑗 and decreasing volumetric flow rate 𝑉𝛼𝑀𝑗, or to infinity with increasing volumetric flow rate 𝑉𝛼𝑀𝑗 and decreasing volumetric flow rate 𝑉𝛽𝑀𝑗. This also means that if the weight fractions of the mixture 𝑀𝑗 approach the binodal, or the composition on the binodal, the flow ratio 𝑓𝑀𝑗 approaches zero (𝑤𝑘,𝑀𝑗 = 𝑤𝛽𝑘,𝑀𝑗) or infinity (𝑤𝑘,𝑀𝑗 = 𝑤𝛼𝑘,𝑀𝑗). Using the given definition of the flow ratio, the following two equations can be obtained:
lim 𝑤𝑘,𝑀𝑗 = 𝑤𝛽𝑘,𝑀𝑗
(10)
lim 𝑤𝑘,𝑀𝑗 = 𝑤𝛼𝑘,𝑀𝑗
(11)
𝑓𝑀𝑗→0
𝑓𝑀𝑗→∞
Equations (10) and (11) show that it is possible to exploit information on the volumetric flow rates of the conjugated phases to calculate compositions on the binodal. In Figure 1 (a), a plot of the weight fraction of the 𝑘-th compound at different mixtures 𝑀𝑗 over the logarithmic flow ratio shows a sigmoidal curve. Following the sigmoidal curve toward its limits (zero and infinity), it is possible to obtain the compositions of the 𝑘-th compound on the binodal. Once the compositions on the binodal are known the ternary phase diagram can be drawn (Figure 1 (b)). The ternary biphasic system consisting of water, acetone, and toluene is taken as an example to demonstrate and validate the proposed procedure and is summarized in Figure 1.
(a)
(b)
Figure 1: Demonstration of the measurement and calculation principle. (a) Weight fractions of the 𝑘-th compound at different mixtures 𝑀𝑗 over the flow ratio 𝑓𝑀𝑗. Shown is a ternary biphasic system consisting of water, acetone, and toluene (WAT). Symbols: circles-water, triangles-acetone, and squares-toluene. (b) Ternary phase diagram of WAT system in weight fractions. Triangles indicate the resulting compositions on the binodal. The initial mixtures are represented by circles. As the components mix, the composition is shown as the dashed line. In Figure 1, it is assumed that there are two pumps. The first pump pumps a homogeneous mixture of toluene and acetone (𝑤𝑇𝑜 = 0.38 and 𝑤𝐴𝑐 = 0.62) at a constant flow rate. The second pump is then switched on and pumps a homogeneous mixture of water and acetone (𝑤𝑊 = 0.53 and
𝑤𝐴𝑐 = 0.47). Both mixtures are mixed on the microchip, separated into two conjugated phases along the microchip channel, and thermodynamic equilibrium is attained, assuming the channel is sufficiently long enough. The volumetric flow rate of the second pump is slowly increased in a stepwise manner. The location of the phase interface is recorded after every change of the volumetric flow rate of the second pump. Each flow ratio results in a heterogeneous mixture with a specific overall composition 𝑤𝑘,𝑀𝑗. Furthermore, as shown in Figure 1 (b), both initial mixtures mix along a straight line that connects the initial mixtures. Therefore, when the volumetric flow rate of the second pump is increased, the mixture composition will change along this straight line toward the second initial mixture composition. The composition on the binodal of the water-rich phase can be accessed from the left side of Figure 1 (a). Hence, the flow ratio converges in the binodal composition of the weight fraction of the 𝑘-th compound at zero as proposed in equation (10). Accordingly, the composition of the toluene-rich phase can be accessed on the right side of Figure 1 (a) (see equation (11)). The resulting binodal compositions are 𝑤𝑎𝑞 𝑊 = 0.503 and 𝑜𝑟𝑔 𝑜𝑟𝑔 𝑤𝑎𝑞 𝐴𝑐 = 0.478 for the aqueous phase, and 𝑤𝑊 = 0.059 and 𝑤𝐴𝑐 = 0.600 organic phase,
respectively. This procedure must be repeated with different initial mixtures to determine other compositions on the binodal and, thus, to determine the complete binodal. 2.1.
Approximation of the flow ratio
In the previous section, we developed a method that utilizes information on the ratio of volumetric flow rates of the conjugated phases (𝑓𝑀𝑗) in a microchannel to calculate compositions on the binodal for a biphasic system. Within this method, it was assumed that the flow rates of the conjugated phases of a mixture 𝑀𝑗 can be measured and, hence, the composition of a heterogeneous, multicomponent biphasic system can be calculated from the mass balance (equation (6)). A non-invasive measurement of the flow rates in equilibrium, however, is not trivial, even though there are several different methods described in literature [29-35]. Many flow measurements are droplet-based and, consequently, inappropriate for the proposed method, which assumes two translucent, laminar, parallel flows separated by an interface. Therefore, in this work, the flow ratio will be estimated as follows. In Figure 2, a microchannel with a biphasic flow of two equilibrated phases and the phase interface are schematically presented.
(a)
(b)
Figure 2: Schematic presentation of a microfluidic channel. The phase interface between two equilibrated liquid phases is given by a black dashed line. The red dashed line is meant to guide the readers eye. (a) The microchannel is observed as seen by the microscope. The flow pattern is parallel and laminar. (b) Cross section of a microchannel. Considering Figure 2, the flow ratio 𝑓𝑀𝑗 defined in equation (7) can be written in terms of the velocities of the phases: 𝑓𝑀𝑗 ≡
𝑉𝛼𝑀𝑗 𝑉𝛽𝑀𝑗
=
𝑣𝛼𝑀𝑗𝐴𝛼𝑀𝑗 𝑣𝛽𝑀𝑗𝐴𝛽𝑀𝑗
=
𝑣𝛼𝑀𝑗𝑙𝛼𝑀𝑗 𝑣𝛽𝑀𝑗𝑙𝛽𝑀𝑗
(12)
where 𝑣𝑀𝑗, 𝐴𝑀𝑗, and 𝑙𝑀𝑗 are the linear velocity, flow cross sectional area, and distance between the interface and the wall of the microchannel for each phase of a mixture 𝑀𝑗, respectively. For moderate differences in density and viscosity, it has been shown that the velocity profile of both phases can be approximated by a velocity profile that one hypothetical fluid would experience when flowing alone at the same flowing conditions [36]. Therefore, the velocity of both phases is assumed to be equal. With this assumption, the flow ratio 𝑓𝑀𝑗 can be approximated by the ratio of the distances between the phase interface and the microchannel walls, which can be determined by visual observation using a microscope. The flow ratio is then given by: 𝑓𝑀𝑗 ≈
𝑙𝛼𝑀𝑗 𝑙𝛽𝑀𝑗
(13)
Equation (13) provides an easier determination of the compositions on the binodal using equation (7), since the phase flow ratio can be easily calculated utilizing an image processing software to determine the distance between the interface and the wall of the microchannel for each phase. 2.2.
Calculation of binodal composition
In Figure 1 (a), the overall composition of the heterogeneous mixture is shown as a function of the flow ratio 𝑓𝑀𝑗. The weight fraction of the 𝑘-th compound in mixture 𝑀𝑗 is a sigmoidal shape similar to the function of the flow ratio that is bound to conditions given by equations (10) and (11). In literature, sigmoidal functions are often used to model bacteria growth [37]. Within this work, a sigmoidal-like function with four coefficients for each compound has been defined and is given in equation (14): 𝑤𝑘,𝑀𝑗 (𝑓𝑀𝑗) =
𝐴𝑘 1 + exp (𝐵𝑘ln (𝑓𝑀𝑗) ― 𝐶𝑘)
+ 𝐷𝑘
(14)
where 𝐴𝑘, 𝐵𝑘, 𝐶𝑘, and 𝐷𝑘 are adjustable parameters for each compound in mixture 𝑀𝑗. The four coefficients of the function are fitted to the measured data to determine compositions on the binodal of the 𝑘-th compound by extrapolating to zero or infinity. To reduce the computational complexity, coefficients of equation (14) are fitted to the experimental data of (𝑁𝑐 ―1) compounds. The weight fraction of the 𝑁𝑐-th compound can be calculated as follows: 𝑁𝑐 ― 1
𝑤𝑚,𝑀𝑗 = 1 ―
∑𝑤
𝑘,𝑀𝑗
(15)
𝑘≠𝑚
The weight fraction of the 𝑁𝑐-th compound in mixture 𝑀𝑗 can be chosen arbitrarily. In the following, the weight fraction of the 𝑁𝑐-th compound in mixture 𝑀𝑗 is acetone. Equation (14) can be extended to account for more asymmetrical trends. This, however, implies using a sophisticated numerical fitting procedure of all adjustable parameters. In addition, adjustable parameters must be sufficiently constrained so that calculated results are physically meaningful. For simplicity, equation (14) has been applied throughout this work. Regarding equation (14), the following physical meaningful constraints can be derived for 𝐴𝑘,𝐵𝑘 > 0. lim 𝑤𝑘,𝑀𝑗 = 𝐴𝑘 + 𝐷𝑘 = 𝑤𝛽𝑘,𝑀𝑗 min (𝑤𝑘,𝑖) ≤ 𝑤𝛽𝑘,𝑀𝑗 ≤ max (𝑤𝑘,𝑖)
(16)
lim 𝑤𝑘,𝑀𝑗 = 𝐷𝑘 = 𝑤𝑘,𝑀𝑗 min (𝑤𝑘,𝑖) ≤ 𝑤𝑘,𝑀𝑗 ≤ max (𝑤𝑘,𝑖)
(17)
𝑓𝑀𝑗→0
𝑓𝑀𝑗→∞
where 𝑤𝑘,𝑖 represents the weight fraction of the 𝑘-th compound in the flow delivered by the 𝑖-th pump. It should be noted that fed flows on the microchip by any pump may be pure solvent or any homogeneous mixture of the solvent system. The minimum and maximum weight fractions of the 𝑘-th compound determine the compositional boundaries. The calculated binodal weight fraction of each compound must obey the constraints given in equations (16) and (17). 3. Experimental section
3.1.
Materials
Acetone (CAS# 67-64-1) was purchase from VWR International GmbH (Darmstadt, Germany) with a purity of at least 99%. Toluene (CAS# 108-88-3) was supplied by Merck KGaA (Darmstadt, Germany) and has a purity of >99.9%. Milli-Q water (18.2 MΩ cm at 25 °C) was used in all conducted experiments. Isopropanol (CAS# 67-63-0) obtained from VWR International GmbH (Darmstadt, Germany) was used to clean the set-up after each experiment. 3.2.
Microfluidic set-up
The microfluidic set-up is schematically shown in Figure 3. The pure solvents, or homogeneous solvent mixtures were placed in the 100 mL reservoirs. Oil-free, pressurized air was used to apply pressure above the surface of a liquid inside the reservoirs by a flow controller (OB1 Base Mk3+, Elveflow, France). Due to this pressure, liquid was pushed through a tube that had been submerged into the liquid toward a flow rate sensor (MINI CORI-FLOW BFS1, Bronkhorst, Netherland). The flow rate sensor was connected to the microchip inlets. The Y-Y microchip made of glass was manufactured by Micronit Microtechnologies B.V. (Netherland). Its channel length, width, and depth were 3 cm, 400 µm, and 50 µm, respectively. The volumetric flow rate was controlled by a feedback loop of the flow rate sensor to the flow controller. The set-up enabled a precise, pulsation-free, and stable volumetric flow rate. The outlets of the microchip were connected to a waste container. The temperature was checked using a thermometer (digital Minithermometer, VWR Collection, Germany) at the outlet. An inverse microscope (Wilovert standard ph 20, Helmut Hund GmbH, Germany) was used to track the phase interface.
Figure 3: Schematic representation of the microfluidic set-up. 3.3.
Experimental procedure
In this work, only pre-equilibrated phases were pumped on the microchip. Therefore, a mixture composition within the biphasic region was selected and prepared in a sealable container. The mixture was thoroughly mixed using a magnetic stirrer for approximately two hours. After that, the mixture was transferred to a separating funnel where it equilibrated for approximately two hours. The room temperature was maintained at 20 °C. Both phases were separated and filled in reservoirs. Compositions of both phases were determined by gas chromatography (GC) using a thermal conductivity detector after each experiment. The GC method is described in section 3.4. The channel inlets 1 and 2 were fed with the water-rich and toluene-rich phases, respectively. The volumetric flow rate of the water-rich phase was held at a constant value. After the microchip was filled with the water-rich phase, the toluene-rich phase was fed to the microchip at a lower initial volumetric flow rate resulting in a small volumetric flow rate ratio which defined the starting point for all experiments. The flow rate of the toluene-rich phase was increased stepwise, which resulted in different locations of the interface. After each stepwise increase of the toluene-rich phase, the volumetric flow rates were held constant for at least two minutes to ensure that the location of the interface did not change further. When the location of the interface remained constant, an image was taken and the volumetric flow rate of the toluene-rich phase was increased again. It should be mentioned that a stable laminar, parallel flow pattern occurs in a certain range of the volumetric
flow rate ratio, and depends on the system properties. Outside of this range, droplet or slug flow patterns were observed. Therefore, as soon as no stable parallel flow was observed, the initially– selected water-rich phase flow rate was increased. The maximum applied volumetric flow rate of the water- and toluene-rich phases were 800 and 1000 µL min−1, respectively. Furthermore, the minimum and maximum ratios of the volumetric flow rates were 0.02 and 32, respectively. Note that these are ratios determined from set volumetric flow rates of the toluene-rich to water-rich phase. 3.4.
GC analysis
A GC from Shimadzu (Nexis GC 2030) with a thermal conductivity detector (TCD) in split mode was used to analyze the phase composition of the samples taken at the end of each experiment from each reservoir. THF was used to dilute the samples. A Capillary column (30 m length, 0.25 mm ID, and 1.4 µm film thickness) obtained from Restek (Rxi®-624Sil MS) was used for the analysis. Helium was used as the carrier gas with a linear velocity of 40 cm s−1. The temperature of the injection port was set to 250 °C at a split-ratio of 50. To separate water and toluene, an oven temperature gradient was applied after an isothermal step at one minute at 35 °C. The oven temperature gradient was linear and increased 31 °C min−1 to 190 °C. The TCD temperature was set to 260 °C. Samples of both equilibrated phases were placed in the two reservoirs (see Figure 3) and taken at the end of each experiment to analyze their compositions by GC. The determined phase compositions were used to calculate the weight fractions of the 𝑘-th compounds for mixtures 𝑀𝑗, according to Equation (6). Note that analyses by GC are unnecessary for the proposed method since pure solvents or mixtures with known compositions are intended to be used as mixtures fed on the microchip and are not the same as the flow ratios approximated by Equation (13). 3.5.
Determination of the flow ratio
The location of the interfacial boundary was determined by using images obtained with a 5 MP camera with a 1/2'' CMOS-sensor (dhs GmbH, Germany) that was mounted on the inverse microscope. The analyses were done post-experimentally by using image processing toolboxes from MATLAB (version R2017b, MathWorks Inc., USA). The locations of the channel boundaries and interface were then used to calculate the flow ratio (equation (13)). In Figure 4, a typical image taken by the camera mounted on the microscope is shown.
(a)
(b)
(c)
Figure 4: Typical image of the micro channel inlet, middle, and outlet before image treatment. As expected, since the two phases are pre-equilibrated before being fed on the microchip, the position of the phase interphase is constant along the channel of the microchip. Images taken by a camera mounted on the microscope were treated so that the flow ratio could be calculated from the ratio of distances of the phase interface to the channel wall (see section 2.1). Before the images were converted into complementary, binary images, all images were first transformed to gray scale and a Gaussian filter was applied. After that, all blobs were removed and a morphological operation of the microchip and phase interphase to a skeleton structure was performed. The developed image treatment allows a direct determination of the phase interface and the channel walls. Due to perturbations at the outlet of the channel, the last location at which the phase interface was determined was at about 60% of the total channel length. For experiments with non-pre-equilibrated phases, the following should be considered when selecting the position for taking images for the determination of the interface position: (i) the system should have reached equilibrium and (ii) the position of the phase interface should not be influenced by the channel outlet. All images were treated according to the proposed image treatment procedure. The flow ratio was then determined by the ratio of the determined lengths of the toluene-rich to the water-rich phase (see Equation (13)). 4. Results and discussion In this work, a novel method based on mass balances was proposed to determine compositions on the binodal utilizing a microfluidic set-up (see Figure 3). The theoretical background of the method was described in section 2. For experimental validation, a WAT system was selected. This system is used as a standard system in liquid-liquid extraction research, and its liquid-liquid equilibrium, including the density, dynamic viscosity, and interfacial tension, has been already extensively measured and documented. Correlations for the calculation of the binodal and phase properties are available in literature [38]. Additionally, to avoid unnecessary discussions about whether or not the length of the microchannel is sufficiently long enough for the biphasic system to reach thermodynamic
equilibrium, pre-equilibrated phases were used to validate the proposed procedure. As already mentioned above, mass transfer is only governed by diffusion in the perpendicular direction of flow. Hence, if both phases are already in thermodynamic equilibrium, mass transfer can be neglected. This experimental procedure has been chosen to focus on the determination of the binodal compositions. Nevertheless, the proposed method is not limited to the use of preequilibrated phases. In case of non pre-equilibrated phases, the length of the microchannel needs to be sufficiently long to guarantee thermodynamic equilibrium between both phases at the outlet of the microchip. A minimum length of the microchannel can be roughly estimated by dividing the product of the mean phase velocity and the squared channel width by the diffusion coefficient of acetone in water or toluene. Four systems were prepared volumetrically as described in section 3.3. Their compositions are given in Table 1. Table 1: Volumetric compositions of conducted experiments. Interfacial tensions were calculated from a correlation from [38].
System
WAT (v/v/v)
[mN m−1]
I
47.5/5/47.5
26.4
II
40/20/40
12.4
III
30/40/30
4.8
IV
20/60/20
0.9
For all mixtures, the ratio of water to toluene was held constant (1:1). Only the acetone content increased from system I to IV. In preliminary experiments, it was found that the standard error of between repeated measurements was sufficiently small. Hence, all experiments were conducted once. Before any experiment was conducted, equilibrated phases were prepared according to the described procedure in section 3.3. Both phases were then fed to the chip. The flow rate of the water-rich phase was kept constant, while the toluene-rich phase was increased stepwise. Under these conditions, a laminar, parallel flow pattern was observed. A biphasic flow and, consequently, the phase interface in a microchannel, was determined by interfacial tension and inertial and viscous forces. To characterize the flow, dimensionless numbers were used, namely Reynolds (inertial/frictional forces), Weber (inertial forces/interfacial tension), and Capillary numbers (frictional forces/interfacial tension). The ranges of the Reynolds, Weber, and Capillary numbers are summarized in Table 2.
Table 2: Ranges of dimensionless numbers, Reynolds, Capillary, and Weber numbers, for all conducted experiments. Compositions of systems I to IV are given in Table 1. No.
Reaq
Reorg
I
9.7–87
II
1.2–8.6
4.5–64.4
III
1.0–12.9
IV
0.9–6.1
Caaq
Caorg
Weaq
Weorg
3.9 10−3–2.9 10−2
0.05–3.8
0.08–3.9
2.0 10−3–1.5 10−2
1.5 10−3–1.9 10−2
0.00–0.1
0.01–1.2
4.8–64.0
5.6 10−3–7.5 10−2
4.9 10−3–5.7 10−2
0.01–1.0
0.02–3.7
3.7–51.8
3.2 10−2–2.2 10−1
3.0 10−2–3.7 10−1
0.03–1.3
0.11–19
19.6–134.5 5.0 10−3–4.4 10−2
The calculated Reynolds numbers indicate a laminar flow, and the Capillary and Weber numbers indicate a parallel flow in the microchannel, which is in accordance to observations for a microchannel with a rectangular cross section [39, 40]. It can be noted that the interfacial tension decreases exponentially with increasing acetone content toward the plait point (𝑤𝑊 = 0.254, 𝑤𝐴𝑐 = 0.637, and 𝑤𝑇 = 0.109 [38]), from systems I to IV (see Table 1). In attaining the plait point, the phase interfacial tension vanishes since the biphasic system collapses to one phase. Due to this fact the proposed method is not capable of measuring the plait point composition. The weight fractions of the 𝑘-th compounds for mixtures 𝑀𝑗 versus the determined flow ratios 𝑓𝑀𝑗 from equation (13), for the four studied systems, are shown in Figure 5.
(a) 47.5/5/47.5
(b) 40/20/40
(c) 30/40/30
(d) 20/60/20
Figure 5: Weight fractions of water and toluene as a function of the flow ratio for systems I-IV listed in Table 1. In Figure 5, the flow ratios 𝑓𝑀𝑗 for each mixture 𝑀𝑗 were determined from images using the approximation given in equation (13) (symbols). The weight fractions of water and toluene as a function of flow ratios were fitted with equation (14). The coefficients of equation (14) were fitted to experimental data following the conditions given in equations (16) and (17). Dashed lines in Figure 5 represent a 95% confidence interval of the fitted coefficients. As demonstrated in Figure 5, the fitted functions are in a good agreement (𝑅2 > 0.99) with the experimental data. This means that equation (14) is reasonable for a phenomenological description of the experimental data. The calculated compositions on the binodal are shown in Figure 6 for the four initial compositions given in Table 1.
Figure 6: Phase diagram of water, acetone, and toluene. The solid line represents binodal compositions calculated from a correlation taken from literature [38]. The circles represent calculated compositions on the binodal by the proposed method and are connected through dashed–dotted lines to indicate the simultaneous calculations of binodal compositions in the water- and toluene-rich phases. (AADW = 0.0017, AADT = 0.0013, AAD = 0.0015) To evaluate the precision of the proposed method, the minimal distance 𝑑𝑚𝑖𝑛 between the experimentally-determined phase composition with the proposed method and the calculated phase composition using the experimentally obtained correlation from Mísek et al. [38] was defined as 𝑁𝐶
𝑑𝑚𝑖𝑛 =
∑(𝑤
𝑒𝑥𝑝 𝑘
2
― 𝑤𝑙𝑖𝑡 𝑘 )
(18)
𝑘
where 𝑤𝑒𝑥𝑝 is the experimentally-determined weight fraction of the 𝑘-th compound. 𝑤𝑙𝑖𝑡 𝑘 𝑘 is the weight fraction of the 𝑘-th compound calculated using the correlation taken from literature [38]. The value of 𝑤𝑙𝑖𝑡 𝑘 was iteratively calculated to minimize equation (18). An absolute average deviation (AAD) between both weight fractions was calculated using equation (19): 𝑚
AAD =
𝑝
𝑐
∑ ∑ ∑ |𝑤
𝑙𝑖𝑡,𝑗 𝑘,𝑙
𝑙 = 1𝑗 = 1𝑘 = 1
― 𝑤𝑒𝑥𝑝,𝑗 𝑘,𝑙 |
(19)
where 𝑐, 𝑝, and 𝑚 represent the number of compounds in the system, phases, and measured 𝑒𝑥𝑝,𝑗 points, respectively. 𝑤𝑙𝑖𝑡,𝑗 𝑘,𝑙 and 𝑤𝑘,𝑙 are literature and experimentally-determined weight fractions
of the 𝑘-th compounds in the 𝑗-th phases of the 𝑙-th data, respectively. Compositions on the binodal in both phases, water- and toluene-rich phases, are in an overall good agreement with literature (AADW = 0.0017, AADT = 0.0013, AAD = 0.0015). Finally, the experimental data were fitted by the following empirical equation that is often used in literature to fit cloud-point titration data of aqueous biphasic systems [11]. 3 𝑤𝑇 = 𝐴exp {𝐵𝑤0.5 𝑊 + 𝐶𝑤𝑊}
(20)
𝐴, 𝐵, and 𝐶 are adjustable parameters that are fitted to the determined binodal compositions. In Figure 7, the fitted binodal is in a good agreement with our experimental data (𝑅2 > 0.99) and data taken from literature [38].
Figure 7: Phase diagram of water, acetone, and toluene. The solid line represents binodal compositions calculated from a correlation taken from literature [38]. The dashed line represents results for the fit of equation (20) to determine binodal compositions. It is shown that the binodal can be accurately described with a few precise measurements. Hence, the proposed method can be used for the determination of the binodal and miscibility gap of multicomponent liquid-liquid systems. 5. Conclusion
In this work, a novel method to determine compositions on the binodal of multicomponent mixtures was theoretically developed and experimentally validated for a ternary system. The proposed method utilized a microfluidic set-up with a pulsation-free pump paired with flow meters controlled by a feedback loop to the pressure unit. This allows the microchip to be fed with a stable and precise volumetric flow rate so that the overall mass on the chip of any compound is known and, furthermore, able to be defined. The set-up reduces the solvent and time consumption. Additionally, it eases handling toxic compounds due to a closed system. The proposed method is based on the assumption of a stable laminar, parallel flow pattern. Therefore, the volumetric flow rates at the inlets of the microchip should be selected to ensure that a stable laminar, parallel flow pattern is obtained in the microchannel. A simple extrapolation procedure has been developed to determine the compositions on the binodal by using the information on the volumetric flow rates and the position of the phase interface of the two equilibrated phases in the microchannel. The overall composition of the system is determined by setting the flow rates of inlet streams consisting of pure component or a homogeneous mixture of the system components that result in changes of the position of the phase interface. An image processing procedure was developed to determine the position of the phase interface in the microchannel by a suitable approximation of the resulting flow ratio. Compositions on the binodal were governed in two steps. First, a sigmoidal function was fitted to overall weight fractions of the 𝑘-th compound at mixture 𝑀𝑗 as a function of approximated flow ratios. After that, the sigmoidal functions were extrapolated to zero and infinity, respectively, to determine the compositions in each respective phase. As an extension of the proposed method, information on the binodal compositions may be used in a subsequent step to calculate the phase equilibrium compositions by measuring an additional property of one of the phases at each set flow rate. Therefore, this method is a promising for the simultaneous determination of the binodal and tie lines. The composition of a liquid-liquid system in equilibrium could be characterized with a few experiments using less solvent than in conventional methods. 6. Acknowledgment This work is part of a cooperation project that is funded by the Bayrische Forschungsstiftung (AZ1285-17) to promote and strengthen the Bavarian position in science and technology. This work was supported by the German Research Foundation (DFG) and the Technical University of Munich within the funding program Open Access Publishing. The authors are thankful the financial support.
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Conflict of interest The authors declare no conflict of interest.
Highlights
A novel microfluidic approach for determination of binodal compositions is proposed
A laminar, parallel flow is observed in the microchannel
Position of the phase interface is captured by an inverse microscope
Determined miscibility gab is in a good agreement with literature data