Rapid Measurement of Concentration-Dependent Viscosity Based on the Imagery of Liquid-Core Cylindrical Lens

Abstract

Viscosity is an inherent frictional characteristic of fluids that enables them to resist flow or deformation, thereby reflecting their flow resistance. It is significantly affected by concentration, but traditional viscosity measurements are limited to discrete concentrations, and multiple experiments are required for different concentrations, so the process is time-consuming. To overcome this limitation, this study presents a “viscosity–diffusion coupling” measurement system using a liquid-core cylindrical lens (LCL) as both the diffusion chamber and imaging element. It captures concentration profiles via focal plane imaging and solves Fick’s second law and Stokes–Einstein relation numerically to determine the viscosity at varying concentrations. Experiments on the viscosity of glycerol solutions (0–50% mass fraction) at three temperatures were conducted and showed strong agreement with literature values. The method enables continuous viscosity measurement across varying concentrations within a single experiment, demonstrating reliability, accuracy, and stability in the rapid assessment of concentration-dependent viscosity.

1. Introduction

Liquid viscosity, as a crucial manifestation of the intermolecular forces within the liquid, stands as one of the key physical properties and technical indicators for characterizing the flow resistance of the liquid, and precise measurement of liquid viscosity holds great significance in a wide range of industrial sectors, including petrochemical, pharmaceutical, metallurgical, and food industries, as well as in scientific research. Consequently, liquid viscosity measurement techniques have consistently received substantial attention [1,2,3,4,5].

Traditional viscosity measurement methods obtain the liquid viscosity by characterizing their flow properties under specific conditions. The Falling Ball Method [6,7,8] determines viscosity by measuring the velocity of a ball descending in a liquid, where resistance is proportional to viscosity. It calculates viscosity through fall time measurement and is valued for its simple setup and efficiency. However, its precision is limited, and ensuring no turbulence during the ball’s fall is crucial for accurate measurements. The rotational viscometer [9,10,11] calculates viscosity by measuring the resistance experienced by a rotating shaft in a liquid, which is proportional to viscosity. It requires determining rotational speed and torque. This method provides high precision and good repeatability. However, it has drawbacks including complex equipment, complicated procedures, and low efficiency. The vibration method [12,13,14] determines viscosity by measuring the damping effect of a vibrating rod immersed in a liquid, where the damping is directly proportional to the viscosity. Viscosity is calculated based on amplitude and frequency data obtained during the vibration process. This method is particularly suitable for high-viscosity liquids and provides relatively high measurement precision. Nevertheless, its application is limited by the complexity of the equipment and the cumbersome operational procedures involved.

In addition, optical measurement methods are favored by researchers due to their non-contact nature and high precision. The optical tweezers technique [15,16,17] uses a focused laser to create an optical trap that captures micro- and nanoparticles. It measures viscosity by analyzing particle motion in the fluid, such as Brownian motion or forced vibration. However, the method requires precise optical components and nanoscale positioning stages. As a result, the system is complex and expensive. The interference method [18,19,20] calculates viscosity by analyzing light interference fringes in fluid films based on lubrication theory and optical interference principles, which enables non-contact, high-precision viscosity measurement, but it is also highly sensitive to environmental disturbances, and the required equipment is complex and expensive. In summary, the viscosity of liquids is significantly affected by factors such as temperature and concentration. The traditional viscosity measurement techniques discussed previously exhibit a common limitation: viscosity assessments are typically limited to discrete concentrations and struggle to capture the complex viscosity–concentration relationship. Multiple concentration measurements require separate experiments, leading to significant experimental effort.

To overcome the limitations previously identified, this study introduces a “viscosity–diffusion coupling” measurement system that employs a custom-designed LCL serving simultaneously as a liquid diffusion chamber and an imaging component. Utilizing focal plane imaging technology, the system captures experimental data reflecting the spatial concentration profile of the diffusing solution within the liquid core at discrete time points [21]. By applying the finite difference method to numerically solve Fick’s second law, the numerical solutions are calibrated to closely match the experimental observations, enabling the determination of the diffusion coefficient D(C) across varying concentrations. Subsequently, integrating the Stokes–Einstein relation allows for the calculation of solution viscosity at different concentrations. In the experiment, this approach effectively measures the viscosity of glycerol solutions within a concentration range of 0–50% by a single experiment at three temperatures (298.15 K, 303.15 K, 307.15 K), and the viscosity measurement deviation can reach 0.05 mPa·s.

2. Calculated Method of Liquid Viscosity

In general solutions, the interactions between solute molecules cause the diffusion coefficient D to depend significantly on the concentration C [22]. The relationship between the diffusion coefficient D(C) and the concentration C can be expressed as

$$D\left(C\right)=D_0\left(1+k_{1}C+k_2{C}^2+k_3{C}^3+\cdots \right)$$

(1)

where k₁, k₂, and k₃ are undetermined coefficients, and D₀ represents the diffusion coefficient of an infinitely dilute solution, measured by transient image analysis method [21].

Fick’s second law describes the law governing the temporal evolution of the concentration profile during the diffusion process [23], and the concentration profile C(x,t) is governed by the one-dimensional diffusion equation (Equation (2)) [24]:

$${\displaystyle \frac{\partial C\left(x,t\right)}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left[D\left(C\right){\displaystyle \frac{\partial C\left(x,t\right)}{\partial x}}\right]$$

(2)

In the numerical calculation, the position of x = 0 is defined as the interface between the two diffusing solutions, with initial concentrations C₁ and C₂, respectively. The diffusion lengths of the two solutions are assumed to be long enough such that the concentrations at both interfaces always remain at the initial values C₁ and C₂. Given that the height of the LCL is L, the initial conditions and boundary conditions are as follows:

$$\left\{\begin{array}{l}t=0\\ L/2>x>0,C=C_1\\ 0>x>-L/2,C=C_2\end{array}\right., \left\{\begin{array}{l}\infty >t>0\\ x=L/2,C=C_1\\ x=-L/2,C=C_2\end{array}\right.$$

(3)

Generally, Equation (2) can be transformed as

$${\displaystyle \frac{\partial C\left(x,t\right)}{\partial t}}=D\left(C\right){\displaystyle \frac{{\partial }^{2}C\left(x,t\right)}{\partial x^2}}+{\displaystyle \frac{\partial D\left(C\right)}{\partial C}}{\left({\displaystyle \frac{\partial C\left(x,t\right)}{\partial x}}\right)}^2$$

(4)

where Equation (4) is solved numerically using the implicit finite difference method [25]; the time step length of the diffusion process is Δt; the spatial domain L along the diffusion direction is discretized into N equally spaced nodes with a grid spacing Δx, respectively; the temporal points t_i = iΔt, where i = 0, 1, 2, …; and spatial positions of the j-th node are defined as x_j = jΔx, where j = 0, 1, 2, …,N. Equation (4) is rewritten as

$${\displaystyle \frac{C_j^{i+1}-C_j^i}{\Delta t}}={\displaystyle \frac{(D_{j+1}^i-D_{j-1}^i)}{2\Delta x}}\times {\displaystyle \frac{(C_{j+1}^{i+1}-C_{j-1}^{i+1})}{2\Delta x}}+D_j^i\times {\displaystyle \frac{(C_{j-1}^{i+1}-2C_j^{i+1}+C_{j+1}^{i+1})}{{(\Delta x)}^2}}$$

(5)

where $D_j^i$ is the difference form of the diffusion coefficient as a function of concentration, assuming $r_{j-1}^i=\Delta t\times D_{j-1}^i/\Delta x^2$, $r_j^i=\Delta t\times D_j^i/\Delta x^2$, and $r_{j+1}^i=\Delta t\times D_{j+1}^i/\Delta x^2$. Equation (5) is transformed as

$$C_j^{i+1}-C_j^i={\displaystyle \frac{r_{j+1}^i-r_{j-1}^i}{4}}\times (C_{j+1}^{i+1}-C_{j-1}^{i+1})+r_j^i\times (C_{j-1}^{i+1}-2C_j^{i+1}+C_{j+1}^{i+1}),$$

(6)

Also, assuming $A_j^i=\left(r_{j+1}^i-r_{j-1}^i\right)/4$ and $B_j^i=r_j^i$, Equation (6) is simplified as

$$(A_j^i-B_j^i)C_{j-1}^{i+1}+(1+2B_j^i)C_j^{i+1}+(-A_j^i-B_j^i)C_{j+1}^{i+1}=C_j^i.$$

(7)

Subject to the initial and boundary conditions (Equation (3)), by employing the Thomas Algorithm [26], the numerical solution of the concentration profile Cⁿ(x,t) at time t can be obtained. Also, the experimental diffusion concentration profile C^e(x,t) is obtained by the measurement platform shown in Figure 1. By changing k₁, k₂, and k₃, a series of Cⁿ(x,t)s are calculated and the standard deviation σ_k can be calculated between the C^e(x,t) and Cⁿ(x,t)s based on Equation (8):

$${\sigma }_k=\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_0^N{(C^n(x,t)-C^e(x,t))}^2}}{N}}}$$

(8)

σ_k (k = 2, …, K) is recalculated after selecting the undetermined coefficients [(k₁)_n, (k₂)_n, (k₃)_n]. Among the K groups of standard deviation values, the minimum value σ_k = (σ_k)_min is identified. When the standard deviation reaches its minimum, the undetermined coefficients are the optimal values [k₁, k₂, k₃]_best, and then these optimal coefficients are substituted into Equation (1) to obtain D(C).

The concentration-dependent viscosity ƞ(C) is solely attributable to the diffusion coefficients, the hydrodynamic radius of the solute molecule, and absolute temperature. According to the Stokes–Einstein relation [27,28], the product of viscosity and diffusion coefficients is directly proportional to the absolute temperature:

$$\eta \left(C\right)={\displaystyle \frac{k_{B}T}{6\pi D\left(C\right)r_s}}$$

(9)

where T is absolute temperature, k_B is the Boltzmann constant, and r_s is the hydrodynamic radius. The empirical relationship describing the viscosity ƞ as an exponential function of concentration C can be derived by Equation (9).

3. Experimental Method for Measuring Liquid Viscosity

3.1. Experimental Setup

The liquid viscosity measurement platform, schematically illustrated in Figure 1, comprises the following optical components: a monochromatic laser (λ = 589 nm) is adopted as the working light source, to prevent damage to the image acquisition system caused by high power, and a neutral density filter is fixed behind the laser. A beam expander, composed of a microscope objective, a pinhole filter, and a collimating lens, and a slit are placed at the back. An LCL is formed by cementing two cylindrical lenses, its central liquid-core region functions as the diffusion chamber for the solution, and the spherical aberration is effectively eliminated through adjustments to the radius of curvature of the cylindrical lenses, with R₁= 32.0 mm, R₂ = 24.0 mm, R₃ = 34.7 mm, and R₄ = 79.5 mm, forming a high-resolution CMOS imaging system (6224 × 4168 pixels, with a pixel size of 3.76 μm).

3.2. Experimental Principle

The LCL was precisely filled with two liquids with distinct refractive indices (n₁ and n₂). Upon contact initiation, the liquids will diffuse with each other, resulting in the formation of a continuous refractive index gradient along the x-axis. When the CMOS sensor is positioned on the focal plane where the liquid with refractive index (n_c) can be sharply imaged, regions associated with other refractive indices exhibit defocused patterns, characterized by a distinct “beam-waist” diffusion image, as shown in Figure 2a. This relationship between the image width W and the localized refractive index arises from the fundamental imaging principles of LCL, as shown in Figure 2b, where the diffused width W and the size of the focal length f satisfy the following relationship [29]:

$${\displaystyle \frac{h}{f}}={\displaystyle \frac{W/2}{\left|f-f_c\right|}}$$

(10)

where the width of the collimated beam is h = 13 mm, limited by a variable-width rectangular slit; f_c is the focal length of the LCL with the refractive index of the liquid n_c; and f is the focal length of the LCL with the refractive index of the liquid n. Therefore, based on the width W(x, t) of the diffusion image at a certain moment, the spatial profile n(x,t) of the refractive index along the x direction is obtained.

4. Results and Discussion

4.1. Experimental Arrangement for Viscosity Measurement

The temperature of the experimental environment was maintained at 298.15 K. A glycerol solution with a mass fraction of 50% was injected using a microsyringe. The solution was then allowed to stand for 5–10 min to mitigate the effects of turbulent flow. Next, the upper liquid (deionized water) was slowly injected using a digital injection pump at a rate of 2.2 mL/min. The CMOS sensor was positioned where the collimated light could be sharply imaged with n = n_c = 1.3443. The diffusion process of the glycerol solution in the deionized water was recorded, with diffusion images captured every 5 min after the process began, and typical experimental diffusion images at different times are shown in Figure 3.

4.2. Experimental Concentration Profile from Diffusion Images

First, deionized water (refractive index n = 1.3327) was injected in the liquid core. The CMOS sensor was adjusted to the position x₁ where the refractive index can be sharply imaged at n = 1.3327, and the imaging system could be calibrated. Then, the position of the CMOS sensor was adjusted to the position x₂, where the liquid refractive index n_c = 1.3443 could be sharply imaged. Second, using a precision electronic balance (accuracy: 0.0001 g), eight sets of different concentrations of glycerol solutions (≤50%) were prepared. Different concentrations of glycerol aqueous solutions were successively injected into the LCL, respectively, and their widths of images were recorded. The experimental results were fitted using the least squares method to establish a relationship between the image width W (mm) and the refractive indices n, as shown in Figure 4a, and the fitting relation is as follows:

$$n=\left\{\begin{array}{c}-0.014W+1.3454, n<n_c,\\ 0.013W+1.3433, n>n_c.\end{array}\right.$$

(11)

Furthermore, as illustrated in Figure 4b, the refractive indices of the glycerol solutions were measured using an Abbe refractometer. A linear regression analysis was conducted for the relationship between solution concentrations (0–50%) and refractive index, yielding a strong correlation coefficient (R = 0.9984). The resulting fitting equation is as follows:

$$C=7.6452\times n-10.182$$

(12)

4.3. Measurement Results of the Viscosity

Taking the experimental diffusion images at t = 214 min and 270 min as examples, binarization was performed to obtain the image width profile W^e(x,t). Based on Equations (11) and (12), the refractive index profile n^e(x,t) and diffusion concentration profile C^e(x,t) along the diffusion direction were calculated, as represented by the blue dots in Figure 5.

Subject to the initial and boundary conditions specified in Equation (3), a series of undetermined coefficients (k₁, k₂, k₃) were introduced to define the functional relationship D(C). In the infinite dilute condition, D(C) is independent of C, D(C) = D₀, Equation (2) has an analytical solution, the D₀ value has been measured by using the transient image analysis method introduced in reference [21], and D₀ = 0.912 × 10⁻⁵ cm²/s. Then, using the finite difference method, the temporal and spatial concentration profile Cⁿ(x,t)s at t = 270 min were calculated. The standard deviation σ_k between the experimental values C^e(x, t) and the computed values Cⁿ(x, t) was calculated. By iteratively adjusting the undetermined coefficients [(k₁)_n, (k₂)_n, (k₃)_n], the standard deviation σ_k (n = 2, 3, …, K) was recalculated for each set of coefficients. Among the n groups of σ_k values, the minimum standard deviation σ_k = (σ_k)_min was identified. The corresponding optimal set of coefficients was determined as [k₁ = (k₁)_best = −1.591, k₂ = (k₂)_best = −1.732, k₃ = (k₃)_best = 3.710]. Based on this optimal coefficients, the best-fit polynomial of the concentration-dependent diffusion coefficient at t = 270 min was obtained as D(C) = 0.912 × 10⁻⁵ (1 − 1.591C − 1.732C² + 3.710C³) cm²/s, as shown in Figure 5, and the calculated concentration profile Cⁿ(x, t) showed good agreement with the experimental profile C^e(x, t). In the experiment, eight diffusion images within the time range of 214 to 323 min were selected for analysis, so the average polynomial of ten times was D(C) = 0.912 × 10⁻⁵ (1 − 1.605C − 1.768C² + 3.644C³) cm²/s. Additionally, the aforementioned experiments were conducted at temperatures of 303 K and 307 K, and the results of these experiments can be summarized as follows, D(C) = 1.020 × 10⁻⁵ (1 − 2.432C + 2.890C² − 2.387C³) cm²/s and D(C) = 1.120 × 10⁻⁵ (1 − 2.372C + 2.742C² − 2.135C³) cm²/s, which are shown in Figure 6. According to the literature [27,28], there is no evidence indicating that glycerol forms stable large clusters at high concentrations. Consequently, its hydrodynamic radius approximately remains invariant with concentration, fulfilling the assumptions of Equation (9). The hydrodynamic radius was r_s = 0.27 nm, measured by the Dynamic light scattering method [30,31]. Using the experimentally obtained values of D(C)s, these data were substituted into Equation (9) to calculate the viscosity values ƞ(C) as a function of concentration. Empirical equations for the viscosity as a function of concentration (Equation (13)) were obtained through fitting at three temperatures. Multiple viscosity values at specific concentrations were selected, as presented in

Table 1. Data of viscosities ƞ (mPa·s) for aqueous glycerol solution from T = (298.15, 303.15, 307.15) K.

C/%	Method Using LCL			Method Using Capillary Tube [32]
	298.15 K	303.15 K	307.15 K	298.15 K	303.15 K
0	0.886	0.806	0.744	0.897	0.803
5	0.968	0.910	0.837
10	1.074	1.029	0.944	1.13	1.04
15	1.211	1.164	1.064
20	1.391	1.321	1.203	1.49	1.32
25	1.626	1.506	1.364
30	1.936	1.730	1.557	2.00	1.76
35	2.346	2.014	1.795
40	2.875	2.395	2.105	2.84	2.48
45	3.520	2.953	2.535
50	4.201	3.880	3.197	4.32	3.79

, and the measured values were compared with the corresponding values reported in the literature [32], revealing a close agreement between them.

$$\eta \left(C\right)=\left\{\begin{array}{c}0.892+1.357C+2.247C^2+16.740C^3\begin{array}{cc}\mathrm{mpa}\cdot \mathrm{s}& \left(298.15 \mathrm{K}\right)\end{array} \\ 0.765+4.257C-15.48C^2+38.332C^3 \begin{array}{cc}\mathrm{mpa}\cdot \mathrm{s}& \left(303.15 \mathrm{K}\right)\end{array}\\ 0.718+3.188C-9.110C^2+24.982C^3 \begin{array}{cc}\mathrm{mpa}\cdot \mathrm{s}& \left(307.15 \mathrm{K}\right)\end{array}\end{array}\right.$$

(13)

In this measurement methodology, the accuracy of experimentally determined concentration values directly influences the precision of viscosity measurements. The primary source of error in concentration determination arises from spherical aberration introduced by the LCL. Within the refractive index range of 1.3327 to 1.3972, the system-induced spherical aberration leads to an average imaging width deviation of δW = 28.23 µm, and this deviation corresponds to a concentration measurement error of δC = 0.30%, which in turn results in a calculated viscosity measurement deviation of δη = 0.05 mPa·s.

5. Conclusions

In summary, the viscosity of liquids is significantly influenced by factors such as temperature and concentration, while traditional viscosity measurement methods are generally restricted to evaluating viscosity at discrete concentrations. Conducting viscosity measurements across multiple concentrations necessitates performing separate experiments for each concentration, thereby imposing a substantial experimental burden. To overcome the limitations previously identified, this study introduces a “viscosity–diffusion coupling” measurement system that employs a custom-designed LCL serving simultaneously as both a liquid diffusion chamber and an imaging component. Utilizing focal plane imaging technology, the system captures the experimental concentration profile of the diffusing solution. By applying the finite difference method to numerically solve Fick’s second law of diffusion, numerical solutions of the concentration profile are calculated to closely match the experimental observations, enabling the determination of the diffusion coefficient D(C) across varying concentrations. Subsequently, integrating the Stokes–Einstein relation allows for the calculation of solution viscosity at different concentrations. In this experiment, the viscosity of glycerol solutions within a concentration range of 0–50% at three temperatures (298.15 K, 303.15 K, 307.15 K) has been measured, with the results exhibiting strong agreement with the values documented in existing literature. This method requires only a single experiment to obtain viscosity values as a function of concentration, thereby offering a novel approach to viscosity measurement. Although the method requires preliminary determination of n(C) and r_s, its benefits—in terms of experimental efficiency, reduced sample consumption, and enhanced data quality—underscore its value as an advanced and strategic technique.