Three-step kinetic model for fisetin dye diffusion into fibroin fibre

Abstract

Background

The study’s relevance is determined by the current desire to reduce the negative environmental impact of the textile industry. The study aims to develop and optimise dyeing processes using natural dyes in the textile industry.

Results

The process of dye transfer from solution to Bombyx mori natural silk fibre can be divided into three successive kinetic stages. The first stage involves the adsorption of dye molecules on the active surface of the fibre, the second, their diffusion deep into the fibre towards its centre, and the third, the uniform distribution of dye molecules along the fibre starting from its centre. It is noticed that diffusion at the third stage slows down significantly, and the third stage lasts much longer than the first and second stages. The analysis of experimental data on dye concentration over time on dyed materials and their comparison with hypothetical data will make it possible to establish time intervals for each stage of the process and diffusion coefficients for each of them.

Conclusion

This study has practical implications as it may contribute to more efficient and sustainable dyeing of textile materials using natural dyes, helping to reduce the negative environmental impact of the textile industry, and contributing to our knowledge of diffusion and dyeing processes.

1 Background

The study of the three-step kinetic model of the diffusion of fisetin dye into fibroin fibres has important practical significance in the textile and materials science industry. Understanding the adsorption, diffusion, and fixation of dye in fibroin fibres enables optimisation of dyeing methods, reduction of dye loss, saving energy and resources, and improving the quality of dyed materials [1]. It also facilitates the development of more sustainable and environmentally friendly dyeing processes, which is relevant in today’s environment where environmental sustainability and efficient use of resources are important [2].

The challenges of investigating a three-step kinetic model for the diffusion of the fisetin dye into fibroin fibres can include difficulties in accurately modelling all steps of the process, including adsorption, diffusion, and fixation, due to a variety of factors such as temperature, humidity, structural features of the fibres, and chemical properties of the dye [3]. It is also important to consider the variability of these parameters in real production conditions, which makes it difficult to create universal models. In addition, attention must be paid to environmental aspects, as many dyeing processes can harm the environment, and the search for more sustainable methods becomes an urgent task [4,5,6].

Akbari et al. [7] raise the important question of the need for a more in-depth analysis of diffusion mechanisms and their effect on mass transfer in the textile industry. However, this study did not analyse the effects of different environmental parameters, such as temperature and humidity, on dye diffusion processes and their effects on mass transfer in the textile industry. This means that there is a need for more research in this area. The study conducted by Hafizov [8] emphasises the importance of determining the diffusion coefficients at different stages of the process to increase the efficiency of the dyeing procedure. However, this study did not include an analysis of the effects of the chemical properties of the dyes and their interaction with fibres on the diffusion and dyeing process, which represents a significant aspect for future studies.

The study by Ding et al. [9] explores the relationship between dye structure and dyeing properties in anhydrous dyeing systems. They found that the presence of cyano groups and smaller steric effect substituents in dye molecules improves molecular stacking, leading to better adsorption but reduced diffusion in polyester fibres. This research helps in optimising dyes for D5 nonaqueous systems, enhancing dyeing efficiency. Fan et al. [10] investigate the diffusion behaviour of disperse dyes in supercritical CO₂ dyeing of polyester fibres. Using confocal Raman microscopy, they observed that dye adsorption increases with pressure and, in most cases, temperature. Their study underscores the role of supercritical CO₂'s high diffusivity in achieving uniform dye distribution early in the dyeing process, providing valuable diffusion coefficient data. Burkinshaw [11] introduces a plasticisation model for dye diffusion, emphasising the importance of macromolecule glass transition and water-induced fibre plasticisation. This model explains the temperature-dependent diffusivity of dyes in water-saturated fibres, offering a new perspective on dye diffusion mechanisms in aqueous dyeing processes.

Aghabalayev and Siming [12] highlight the potential of applying the results of the study to developing more environmentally friendly and efficient dyeing methods in the textile industry. However, this study does not focus on the long-term environmental implications of these dyeing methods and their impact on environmental sustainability. Azanaw et al. [13] emphasise that wastewater from textile industries is classified as the most polluting of all industrial sectors, considering both the amount of waste generated and its composition. In addition, the increasing demand for textile products and the proportionate increase in their production, as well as the use of synthetic dyes, have combined to make wastewater from dyeing industries one of the significant sources of serious pollution problems today. Szweda et al. [14] raise an important problem of the methodology of comparison of experimental data and model calculations to determine the time intervals and diffusion coefficients. However, this study does not include an analysis of the effect of different types of dyes on the diffusion and staining process, which represents an essential aspect for subsequent studies.

The aim of this research is to develop and optimise a three-step kinetic model for the diffusion of fisetin dye into fibroin fibres, enhancing dyeing efficiency and sustainability in the textile industry. The research question of this study is centred on understanding and optimising the dyeing process to achieve greater efficiency and sustainability. Specifically, the research question is: “How can the three-step kinetic model be optimised to enhance the efficiency and sustainability of natural dye diffusion into fibroin fibres, considering the impact of environmental parameters and dye-fibre interactions on each stage of the dyeing process?” By addressing this question, the study aims to contribute to the development of more sustainable and efficient dyeing methods in the textile industry, reducing environmental impact and improving resource utilisation.

2 Methods

The structural protein fibroin, which is a major component of natural silk fibres, is an amphoteric polyelectrolyte capable of interacting with both acids and bases. The isoelectric range of fibroin is between 3.5 and 4, and its exchange capacity (in terms of acids) is about 0.2 to 0.3 mol/kg. Fibroin exhibits both hydrophilic and hydrophobic characteristics, but with a clear predominance of hydrophobicity. Its isoelectric point is pI = 4.2, making it insoluble in water.

Due to the structural nature of the fibroin biopolymer, the process of dye diffusion in it takes about 0.5–2 h at temperatures up to 373 K. Fibroin obtained from the fibres of the silk cocoon of the silkworm Bombyx mori has a high specific surface area, and the diameter of the fibres is 14–20 µm. The amorphous part of the fibroin is an organised medium with mobile ends and regions of transient molecules, similar to a viscous fluid. However, the flexible framework limits the possible shapes of oscillating cavities and cracks in the fibroin microfibre structure.

Fisetin (also known as 3,3’,4’,7-tetrahydroxyflavone; C₁₅H₁₀O₆; melting point 330 °C) is a polyphenolic flavonoid that belongs to the class of plant flavanols. This natural phytochemical is found in pigmented fruits and vegetables and has shown beneficial properties, including antioxidant and anti-apoptotic effects in certain forms of malignant tumours. Fisetin is also used as a yellow colouring agent.

The base yellow dye was obtained from the wood of the smoky tree. Cotinus coggygria Scop. is a dark yellow crystalline powder that is readily soluble in methanol and ethanol. Fisetin, which is an antioxidant flavonol, was first extracted in the second half of the nineteenth century to the first half of the twentieth century from the plant Cotinus coggygria Scop. Nowadays, it has found wide application in the production of kelagai—women’s shawls are woven from 7 to 9 unspun threads of mulberry silkworm cocoons. There are many patents and scientific articles describing the process of extracting fisetin from Cotinus coggygria Scop.

The main feature of this method is the fact that the microlevels of diffusion, which represent the process of distribution of fisetin molecules in microfibres, are determined by the characteristics of the medium at the macrolevel, such as density, temperature, and concentration. To study the kinetics of this process, a principle of chemical kinetics known as the “limiting stage principle” was used. In this case, the diffusion process is divided into three consecutive stages, and the diffusion rate determines the overall rate of the whole process.

The basis of the mathematical part of this study is a system of differential kinetic equations, which determine the distribution functions of particles at a particular stage at given velocities. The migration process based on the diffusion rate can be divided into four stages:

• 1. Migration of dye molecules from the outer solution to the fibre surface.

• 2. Motion through a liquid layer close to the liquid–solid boundary.

• 3. Movement through a layer of adsorbed material on a surface.

• 4. Half a metre-deep penetration into the pores or structure of the material. This stage should be divided into two sub-stages: the first, in which the penetrating dye molecules reach the centre of the fibre, and the second, in which a uniform distribution of molecules throughout the fibre has not yet been achieved. This is explained by the change in the concentration gradient of these molecules as they penetrate the fibre and reach its centre, resulting in a slowing down of their speed of movement.

The process of the migration of dye molecules inside the fibre was divided into three stages. In the first stage, the dye molecules are transferred to the active surface of the fibre. In the second stage, the molecules penetrate through the surface-adsorbed layer of dye molecules and reach the central part of the yarn. In the third step, the dye molecules are evenly distributed throughout the entire volume of the fibre. Based on the kinetic properties of fisetin molecules inside fibroin fibres, a working solution was prepared in which 20 g of fisetin was dissolved in 4 L of distilled water with a certain concentration of $C_{\infty } = 0.01747 {\frac{{{\text{mol}}}}{{\text{l}}}}$.

Two solutions of fisetin in distilled water were prepared for an experimental study of the dependence of the relative concentration of fisetin molecules on the staining time: the basic solution and the working solution. Each of the 4-L solutions contained 20 g of dissolved fisetin powder. Two VT5 liquid thermostats with baths made of solid metal a volume of 5 L and a depth of 200 mm were used for the experiment. The samples under study were 5 batches of natural silk fibres (threads from cocoons) weighing 100 g each. These fibres were purified from sericin, fat, and waxy substances using a Grundig filter by incubation in a boiling solution of sodium carbonate and sodium hydrogen carbonate, then extracted with petroleum ether and ethanol, and finally dried until a constant weight was reached. The samples were then placed in working and control solutions. The staining process was carried out under constant stirring to ensure uniform coverage of the adsorbent surface and to avoid an uneven distribution of fisetin throughout the solution.

In conducting the experiments, the author used techniques substantiated by eminent experimentalists in the field:

• Solution of diffusion kinetics problems by the simplest possible methods, but at the same time giving nontrivial results, allowing to solve the problem completely without resorting to numerical methods;

• Solution of problems for baths of finite volume and diffusion into a flat sheet;

• Solution of problems for an infinite cylinder.

The solution of kinetic differential equations is usually represented as a power series. The complexity of such a solution depends on how much the dye concentration in the dye bath changes during the dyeing process. That is, during the dyeing period, the dye concentration in the dye bath either changes or remains constant. According to Carman and Haul [15], if the solution is well mixed, the concentration of dye in it depends only on time, and the presence of such a limited volume of solution is experimentally justified. In practice, the first option is implemented, when a small amount of dye material is treated with a large volume of dye solution and a small amount of dye is absorbed by the fibres. The author applied the first technique and determined the rate of penetration of dye molecules into the sample by observing the rate of change of dye concentration in the solution.

Measurements were taken using the photometric method. Standard rectangular cuvettes with external dimensions of 12.5 mm × 45 mm × 12.5 mm, a working volume of 3.5 mL, and an optical path length of 9.5 mm were used for measurements in the ultraviolet (UV) and visible range, glass type – Q: Spectrosil® quartz glass for far UV, for wavelengths from 190 to 2700 nm. At the end of each minute, a 40 ml sample was taken from the working dye solution and placed in ten cuvettes. Each time, 40 ml of dye solution was taken from the control solution and added to the working solution to maintain a stable volume of active solution. The spectral transmission coefficients of the fisetin solution were measured at a wavelength of 313 nm using the 752N Plus UV–VIS spectrophotometer, User Manual, which operates in the spectral range of 100–1000 nm.

To compensate for the loss of water caused by evaporation during the boiling of the dyeing solution in the working and control baths, it is necessary to add a specific amount of hot distilled water to the bath.

The measurements obtained were processed to determine the relationship between $\frac{{ C_{t} }}{{C_{\infty } }} = f\left( t \right)$ and t, where Δt represents the interval, which lasts approximately one minute. The experiments were performed at 373 K for each point on the graph. Ten measurements were taken (n = 10) for each point on the graph, and the standard deviation ($\widetilde{ \sigma )}$ was calculated using formula (1):

$$\tilde{\sigma } = \sqrt {\frac{{\left\{ {\mathop \sum \nolimits_{i = 1}^{n} \left[ {\left( {\frac{{C_{t} }}{{C_{\infty } }}} \right)^{ \sim } - \left( {\frac{{C_{t} }}{{C_{\infty } }}} \right)_{i} } \right]^{2} } \right\}}}{{\left[ {n\left( {n - 1} \right)} \right]}}} ,$$

(1)

where $\left( {\frac{{C_{t} }}{{C_{\infty } }}} \right)^{ \sim }$—arithmetic mean of measurement results; $\left( {\frac{{C_{t} }}{{C_{\infty } }}} \right)_{i}$—measurement result.

The standard level (as in most general physics laboratory work) was chosen as the confidence level, and using Student’s table, where the coefficient $t_{\alpha n} = 2.26$ depends on α and n, the confidence interval for the random error (error associated with repeated measurements) was determined (2) and (3):

$$\Delta \left( {\frac{{C_{t} }}{{C_{\infty } }}} \right)^{ \sim } = t_{\alpha n} \cdot \tilde{\sigma } = 2.26 \cdot \tilde{\sigma },$$

(2)

$$\left( {\frac{{C_{t} }}{{C_{\infty } }}} \right)^{ - } = \left( {\frac{{C_{t} }}{{C_{\infty } }}} \right)^{ \sim } \pm \Delta \left( {\frac{{C_{t} }}{{C_{\infty } }}} \right)^{ \sim } .$$

(3)

In this section, the main methods for investigating a three-step kinetic model for the diffusion of the dye fisetin into fibroin fibre were presented, providing a deeper understanding of the dye adsorption, diffusion, and fixation processes.

In this study, external factors that could have influenced the results were carefully controlled to minimise bias and confounding variables. Specifically, the experiments were conducted in a controlled environment where temperature and humidity were consistently monitored and maintained within specified ranges. This ensured that any variations in these external conditions did not skew the outcomes. Additionally, while the primary focus of the study was on natural dyes, synthetic dyes were also included to provide a more comprehensive understanding of the diffusion processes. The selection of dyeing processes and materials was conducted objectively to avoid any bias. This was achieved by following standardised protocols and using materials and methods that are widely accepted in the field. The criteria for selection were based on their relevance and suitability for the study, ensuring that the results are accurate and credible.

To ensure the consistency and reliability of the results, a consistent and high-quality source of natural dyes was used throughout the experiments. The dyes were sourced from reputable suppliers, and their quality was verified through preliminary testing before being used in the main study. Furthermore, the impact of critical factors such as temperature and pH on the dyeing process was thoroughly accounted for. The experiments were designed to include a range of temperature and pH conditions, and their effects on dye diffusion were systematically studied. Consistent and controlled conditions were maintained throughout the dyeing experiments to ensure the accuracy and reproducibility of the results. All variables, including dye concentration, dyeing time, and mechanical agitation, were standardised across all experiments. This rigorous control of experimental conditions ensured that the observed results were due to the inherent properties of the dye and fibre interaction rather than external or uncontrolled factors.

3 Results

The diffusion coefficient of dye diffusion within a fibre is of high importance to dyeing specialists because it determines the speed of the dyeing process. On the other hand, the measurement of the diffusion coefficient of small molecules in the polymer matrix is sometimes used to detect changes in the polymer microstructure and to determine phase transitions, since the value of the diffusion coefficient strongly depends on the mobility of segments of polymer molecules. The results of measurements obtained using formulas (1–3) are given in Table 1.

Table 1 Dependence of the relative concentration of fisetin molecules penetrating fibroin fibres on dyeing time. Source: compiled by the authors

Based on the presented data from Table 1, we plotted the experimental dependence of the relative concentration of fisetin molecules ($\frac{{C_{t} }}{{C_{\infty } }}$) in fibroin fibres on the staining time t at a constant temperature of 373 K and presented it in Fig. 1.

Fig. 1

The graph shows the change in the relative concentration of fisetin molecules penetrating the fibroin fibre as a function of staining time at 373 K

This graph shows that the dependence of the relative concentration of fisetin molecules in fibroin fibres on the dyeing time at constant temperature (isothermal diffusion) has the form of a curve with saturation.

3.1 Stages of the first and second diffusion of dye molecules inside the fibre (cylinder)

During the adsorption of adsorbate molecules on the adsorbent, the first stage, which determines further diffusion kinetics, occurs almost instantaneously, which makes its observation difficult [16]. In the dyeing process, it is not possible to distinguish between the first and second stages in which the dye molecules reach the centre of the fibre. Therefore, this study also considers the first and second stages of diffusion as a whole, as well as the third stage separately.

The developed kinetic model gives a mathematical description of the kinetics of the first and second stages of the process jointly and of the third stage. Diffusion is a three-dimensional process. However, the work of Stokes [17] is used for assumptions of limiting the concentration gradient in one direction only, i.e. the diffusion problem is simplified to one-dimensional. In the case of diffusion, a certain amount of substance (adsorbate) (or mass) moves through a certain area of the active surface of the adsorbent during the time interval t, which is along the normal direction $\overrightarrow { x}$, along which the concentration of the substance changes. This amount of substance (or mass) is proportional to the concentration gradient (dC/dx), area, and time of the process’s continuation, according to Fick’s first law (4):

$$\Delta m = J\Delta S\Delta t = - D\left( {\frac{{{\text{d}}C}}{{{\text{d}}x}}} \right)_{T} \Delta S\Delta t,$$

(4)

where $J = - D\left( {\frac{{{\text{d}}C}}{{{\text{d}}x}}} \right)_{T}$—penetration flux density (measures the amount of substance that will pass through a unit area during a unit time interval) is in the following units $\frac{{\left( {{\text{mol}} \cdot {\text{s}}} \right)}}{{{\text{cm}}^{{3}} }}$; $D$—diffusion coefficient, directly proportional to $\overline{\mu }$—average speeds $\overline{\lambda }$—average particle path length (5):

$$D = \frac{{\overline{\mu }}}{3}\overline{\lambda }$$

(5)

Since the flux of matter tends to eliminate the uneven distribution of matter in the system, D measures the rate at which the system can equalise the concentration difference under given conditions. This speed $\overline{\mu }$, in turn, depends on the microscopic parameters of the system, which characterise the thermal mobility of macromolecules in the diffusion medium and diffusing particles. The negative sign in Eq. (4) arises because the particles move in the direction of decreasing concentration. Substituting Eq. (5) into Eq. (4), we obtain the following semi-equal Eq. (6):

$$J = - \frac{{\overline{\mu } \cdot \overline{\lambda }}}{3}\left( {\frac{{{\text{d}}C}}{{{\text{d}}x}}} \right)_{T} .$$

(6)

The amount of substance ${\text{d}}Q_{x}$ (e.g. fisetin molecules) penetrating a fibroin fibre through its outer surface $S$, in the longitudinal direction of the fibre, within an extremely short time ${\text{d}}t$ can be expressed by the following Eq. (7):

$${\text{d}}Q_{x} = JSm_{0} {\text{d}}t,$$

(7)

where in this case $m_{0}$—fisetin molecule mass.

At a constant temperature of the dye solution, from Eqs. (4), (5), (6) and (7), let us determine the quantities of substance (8):

$${\text{d}}Q_{x} = D\left( {\frac{{{\text{d}}C}}{{{\text{d}}x}}} \right)_{T} Sm_{0} {\text{d}}t.$$

(8)

By using Eq. (8), we developed a kinetic model for the diffusion of fisetin molecules in fibroin fibres. We assumed that at the first and second stages of the dyeing process, the dye concentration along the fibre thickness varies linearly, and this made it possible to formulate the following equality:

$$\frac{{{\text{d}}C}}{{{\text{d}}x}} = \frac{{C_{\infty } }}{x},$$

(9)

Equation (9) defines the maximum steady-state dye concentration $C_{\infty }$ that the spontaneous diffusion process can achieve in the sample.

Figure 2 shows a three-dimensional kinetic model representing the dye distributions inside the fibre. To study the diffusion of fisetin molecules inside the fibre, a three-dimensional physical–kinetic model describing the pigment distribution in the first and second stages of dyeing was used. A cross section of the fibre of unit length h = 1 is considered, and the diffusion process is represented on a concentration scale. Consequently, this cross section of a fibre of length one unit has a certain volume: $V = \pi r_{0}^{2} h$ and side surface area: $S = 2\pi r_{0} h$.

Fig. 2

Three-dimensional kinetic model of the dye distributions in fibre

In this model, the position $x_{1}$ represents the point marking the end of the first stage (the adsorption process) and the beginning of the second stage. In this context, the change in concentration as the dye moves through the lateral region over time ${\text{ d}}$ t will be ${\text{d}}C_{t}^{I\& II}$. At these steps, the number of dye molecules penetrating the direction $\vec{x}$ is equal to (10):

$${\text{d}}Q_{x}^{I\& II} = \pi r_{0}^{2} m_{0} {\text{d}}C_{t}^{I\& II} .$$

(10)

Adding (8) to (10) and considering that $h = 1$, we obtain the following Eqs. (11, 12—semi-equal):

$$\pi r_{0}^{2} m_{0} {\text{d}}C_{t}^{I\& II} = D\left( {\frac{{{\text{d}}C_{\infty } }}{{{\text{d}}x}}} \right)Sm_{0} {\text{d}}t,$$

(11)

$$\pi r_{0}^{2} m_{0} {\text{d}}C_{t}^{I\& II} = D\left( {\frac{{{\text{d}}C_{\infty } }}{{{\text{d}}x}}} \right)2\pi r_{0} m_{0} {\text{d}}t.$$

(12)

Considering that the vector radius of fibre $\vec{r}_{0}$ and the direction of motion of the dye molecules $\vec{x}$ are opposite to each other, following Eq. (9), it is possible to substitute for the gradient $\frac{{{\text{d}}C_{\infty } }}{{{\text{d}}x}}$ we can take $\frac{{C_{\infty } }}{x}$, and formulate expression (13) as follows:

$$\frac{{{\text{d}}C_{t}^{I\& II} }}{{C_{\infty } }} = \frac{2}{x}\frac{D}{{r_{0} }}{\text{d}}t.$$

(13)

The last obtained expression (13) is a differential equation to describe the colouring kinetics at the first and second stages of the process. To solve it, we start by integrating the semi-equal expression (14):

$$Q_{t}^{I\& II} = \pi r_{0}^{2} m_{0} C_{t}^{I\& II} + c_{1} ,$$

(14)

where $c_{1}$—integral constant.

At the beginning of the process, we assume that before the initial moment of contact between the dye molecules and the fibre $\left( {t = 0} \right)$, the thread has no dye molecules: $Q_{x} = 0$ and, respectively, $c_{1} = 0$. Therefore, we have a semi-equal Eq. (15):

$$Q_{t}^{I\& II} = \pi r_{0}^{2} m_{0} C_{t}^{I\& II} .$$

(15)

From the beginning to the end of stages I and II of the process, i.e. at the value of $x$ from 0 to $r_{0} ,$ the volume of the cylinder without a trimmed cone varies from 0 to $\left( \frac{2}{3} \right)\pi r_{0}^{2} C_{\infty }$. Considering that the change in concentration as the dye molecules move through the lateral region over time $dt$ will be equal to ${\text{d}}C_{t}^{I\& II}$, and the number of dye molecules penetrating at the first and second stages in the direction of $\vec{x}$ will be semi-equal Eq. (16):

$$Q_{x}^{I\& II} = \pi r_{0}^{2} m_{0} C_{x}^{I\& II} = V_{x}^{I\& II} m_{0} C_{\infty } ,$$

(16)

where (17, 18):

$$V_{x}^{I\& II} = V_{{{\text{cylinder}}}} - V_{{\text{trimmed cone}}} = \pi \left( {r_{0} x - \frac{{x^{2} }}{3}} \right),$$

(17)

$$Q_{x}^{I\& II} = \pi m_{0} C_{\infty } \left( {r_{0} x - \frac{{x^{2} }}{3}} \right).$$

(18)

According to expressions (15, 16), we can conclude that (19—semi-equal equation, 20—semi-equal equation):

$$\left( {r_{0} x - \frac{{x^{2} }}{3}} \right)C_{\infty } = r_{0}^{2} C_{t}^{I\& II} ,$$

(19)

$$x^{2} - 3r_{0} x + 3\frac{{C_{t}^{I\& II} }}{{C_{\infty } }}r_{0}^{2} = 0.$$

(20)

When the diffusing particle reaches the centre of the fibre, $x = r_{0}$ and in this case, $C_{t}^{I\& II}$ reaches its maximum value (21, 22):

$$r_{0}^{2} - 3r_{0}^{2} + 3\frac{{C_{t}^{I\& II} }}{{C_{\infty } }}r_{0}^{2} = 0,$$

(21)

$$C_{{{\text{max}}}}^{I\& II} = \frac{2}{3}C_{\infty } .$$

(22)

Equation (20) is a quadratic equation depending on the variable $x$, and, since $C_{\infty } > C_{t}^{I\& II}$, it has two real number roots (23):

$$x = 1.5r_{0} \left( {1 \pm \sqrt {1 - \frac{4}{3}\frac{{C_{t}^{I\& II} }}{{C_{\infty } }}} } \right).$$

(23)

However, $x \le r_{0}$. Therefore, according to (22) we choose the following solution (24):

$$x = 1.5r_{0} \left( {1 - \sqrt {1 - \frac{4}{3}\frac{{C_{t}^{I\& II} }}{{C_{\infty } }}} } \right).$$

(24)

Substituting (24) into Eq. (13), we obtain the following Eq. (25):

$$\left( {1 - \sqrt {1 - \frac{4}{3} \cdot \frac{{C_{t}^{I\& II} }}{{C_{\infty } }}} } \right)\frac{{{\text{d}}C_{t}^{I\& II} }}{{C_{\infty } }} = \frac{4}{3} \cdot \frac{D}{{r_{0}^{2} }}{\text{d}}t.$$

(25)

After performing the integration of the differential Eq. (25), the following equation involving integrals from irrational functions is obtained (26):

$$\sqrt {\left( {1 - \frac{4}{3} \cdot \frac{{C_{t}^{I\& II} }}{{C_{\infty } }}} \right)^{3} } + 2\frac{{C_{t}^{I\& II} }}{{C_{\infty } }} = \frac{8}{3} \cdot \frac{D}{{r_{0}^{2} }}t + c_{2} .$$

(26)

To calculate integration constant $c_{2}$ it was assumed that the initial point of the second stage of the diffusion process $t^{II}$ coincides with the end point of the first stage, i.e. $t^{I}$. Consequently, we can simplify the equation to $\frac{{C_{t}^{I} }}{{C_{\infty } }}$, which means both equations $\frac{{C_{t}^{II} }}{{C_{\infty } }}$ and $\frac{{C_{t}^{I} }}{{C_{\infty } }}$ will be equal (27):

$$\frac{{C_{t}^{II} }}{{C_{\infty } }} = \frac{{C_{t}^{I} }}{{C_{\infty } }}.$$

(27)

Equations (22–27) are semi-equal and involve complex relationships and approximations.

Substituting equality (27) into Eq. (26) for the beginning of the second stage $\left( {t^{II} = 0} \right)$, the following equation will be true (28):

$$c_{2} = \sqrt {\left( {1 - \frac{4}{3} \cdot \frac{{C_{t\prime 0}^{I} }}{{C_{\infty } }}} \right)^{3} } + 2\frac{{C_{t\prime = 0}^{I} }}{{C_{\infty } }}.$$

(28)

where $C_{t\prime = 0}^{I}$ represents the concentration of dye molecules inside the fibre at the end of the first stage of the process.

If we substitute this expression $c_{2}$ into Eq. (26), we obtain the following semi-equal Eq. (29) at the end of the first stage of the diffusion process:

$$\sqrt {\left( {1 - \frac{4}{3} \cdot \frac{{C_{t}^{II} }}{{C_{\infty } }}} \right)^{3} } + 2\frac{{C_{t}^{II} }}{{C_{\infty } }} = \frac{8}{3} \cdot \frac{D}{{r_{0}^{2} }}t + \sqrt {\left( {1 - \frac{4}{3} \cdot \frac{{C_{t\prime 0}^{I} }}{{C_{\infty } }}} \right)^{3} } + 2\frac{{C_{t\prime = 0}^{I} }}{{C_{\infty } }}.$$

(29)

After simplification (30):

$$\frac{{C_{t}^{II} }}{{C_{\infty } }} = \lambda ;\frac{Dt}{{r_{0}^{2} }} = \beta \sqrt {\left( {1 - \frac{4}{3} \cdot \frac{{C_{t\prime = 0}^{I} }}{{C_{\infty } }}} \right)^{3} } + 2\frac{{C_{t\prime = 0}^{I} }}{{C_{\infty } }} = \chi .$$

(30)

Substituting them into (29), we obtain the following Eqs. (31, 32):

$$\sqrt {\left( {1 - \frac{4}{3} \cdot \lambda } \right)^{3} } + 2\lambda = \frac{8}{3}\beta + \chi ,$$

(31)

$$\lambda^{3} - \frac{9}{16}\lambda^{2} + \frac{27}{{16}}\left( {1 - \frac{8}{3}\beta - \chi } \right)\lambda + \left( {3\beta^{2} + \frac{9}{4}\beta \chi + \frac{27}{{64}}\chi^{2} - \frac{27}{{64}}} \right) = 0.$$

(32)

And semi-equal Eq. (32) is reduced to (33):

$$a\lambda^{3} + b\lambda^{2} + c\lambda + d = 0,$$

(33)

where (34):

$$a = 1,b = - \frac{9}{16},c = \frac{27}{{16}}\left( {1 - \frac{8}{3}\beta - \chi } \right),d = 3\beta^{2} + \frac{9}{4}\beta \chi + \frac{27}{{64}}\chi^{2} - \frac{27}{{64}}$$

(34)

At the beginning of the first stage of the diffusion process, i.e. $t^{I} = 0$, dye concentration in the fibre is zero ($C_{t}^{I} = 0$). Therefore, from expression (34) we determine that $\chi = 1$. In this case (35):

$$a = 1;b = - \frac{9}{16};c = - \frac{9}{16}\beta ;{\text{and}}\;d = 3\beta^{2} + \frac{9}{4}\beta$$

(35)

Replacing $\lambda = y - \left( \frac{b}{3} \right)$ in Eq. (33) [18], the following can be remade (36, 37—semi-equal):

$$\left( {y - \frac{b}{3}} \right)^{3} + b\left( {y - \frac{b}{3}} \right)^{2} + c\left( {y - \frac{b}{3}} \right) + d = 0,$$

(36)

$$y^{3} + \left( {c - \frac{{b^{2} }}{3}} \right)y + c\left( {\frac{{2b^{2} }}{27} - \frac{bc}{3} + d} \right) = 0.$$

(37)

Now, by adding these $p = c - \frac{{b^{2} }}{3}$; $q = \frac{{2b^{2} }}{27} - \frac{bc}{3} + d$, we obtain the normalised cubic Eq. (37) in the canonical form (38):

$$y^{3} + py + q = 0.$$

(38)

If $b = - \frac{9}{16}$; $c = - \frac{9}{16}\beta$ and $d = 3\beta^{2} + \frac{9}{4}\beta$, as well as (39, 40—semi-equal):

$$p = c - \frac{{b^{2} }}{3} = \frac{{3c - b^{2} }}{3} = - 4.5\beta - 0.1057,$$

(39)

$$q = \frac{{2b^{3} }}{27} - \frac{bc}{3} + d = 3\beta^{2} + 1.40625\beta - 0.0132.$$

(40)

Let us consider the cubic polynomial $\lambda^{3} + b\lambda^{2} + c\lambda + d = 0$, solving the given cubic Eq. (33), assuming that $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ are its roots. Irving [19] indicates that the value $\delta$ defines initial conditions for the cubic polynomial:

$$\delta = \left( {\lambda_{1} - \lambda_{2} } \right)^{2} \left( {\lambda_{1} - \lambda_{3} } \right)^{2} \left( {\lambda_{3} - \lambda_{2} } \right)^{2} = - 4p^{3} - 27q^{2} .$$

(41)

If $\delta < 0$, a cubic polynomial has one real root and two nonlinear complex-conjugate roots. If $\delta = 0$, then all roots of Eq. (38) are real and at least two of them are equal. At $\delta > 0$, the polynomial has three different real roots. Calculations using expressions (38–40) show that for example $\delta$ it has a negative sign. Therefore, as suggested by Lestari et al. [20], to find the root of the reduced cubic polynomial $\delta^{3} + b\delta^{2} + c\delta + d$ in the first case ($\delta < 0$) used a well-known solution called Cardano’s formula. That is, Eqs. (38) have one real root (42):

$$y_{1} = A + B,$$

(42)

and two assumed roots (43):

$$y_{2,3} = - \frac{A + B}{2} \pm i\frac{A - B}{2}\sqrt 3 ,$$

(43)

where (44, 45):

$$A = \sqrt[3]{{ - \frac{q}{2} + \sqrt \Delta }},B = \sqrt[3]{{ - \frac{q}{2} - \sqrt \Delta }}\;{\text{and}}\;\Delta = \left( \frac{p}{3} \right)^{3} + \left( \frac{q}{2} \right)^{2} = 2.25\beta^{4} - 1.2657\beta^{3} + 0.317\beta^{2} - 0.0148\beta$$

(44)

$$y_{1} = A + B = \sqrt[3]{{ - \frac{q}{2} + \sqrt \Delta }} + \sqrt[3]{{ - \frac{q}{2} - \sqrt \Delta }}.$$

(45)

Considering $\beta = \frac{Dt}{{r_{0}^{2} }}$ and $\beta > 0$, Eq. (38) sets $\Delta > 0$. Employing expression $\left( {\frac{{C_{t}^{II} }}{{C_{\infty } }}} \right) = \lambda$ (34) and replacing the expressions $\lambda = y - \left( \frac{b}{3} \right)$, as suggested by Bardell [18], the real root should be used as equality (46):

$$\lambda = \sqrt[3]{{ - \frac{q}{2} + \sqrt \Delta }} + \sqrt[3]{{ - \frac{q}{2} - \sqrt \Delta }} - \frac{b}{3} .$$

(46)

Thus, the dependences of the relative concentration of fisetin molecules in the inner part of the fibroin filament—in the second stage of diffusion, have formulae (47):

$$\frac{{C_{t}^{II} }}{{C_{\infty } }} = \sqrt[3]{{ - \frac{q}{2} + \sqrt \Delta }} + \sqrt[3]{{ - \frac{q}{2} - \sqrt \Delta }} + 0.1845.$$

(47)

The identified mathematical Eq. (47) is a second-order kinetics equation. In the compiled Table 2, the calculated value of Eq. (47) is used. In this case, the value of the function corresponds to any value of the argument in the area of definition $\beta$. When the minimum value reaches $\beta$ of the second stage of diffusion, it turns out that below the estimate of $\beta = 0.0592$ discriminant $\sqrt \Delta$ of the cubic equation, defined by formula (49), becomes a negative number. In this case, Eq. (47) has an irrational root.

Table 2 Calculated values of a function (51) with arbitrary argument value from the definition area.

The minus sign in front of the relative adsorbate concentration values is indicated because the particles move in the direction of decreasing concentration.

In addition, the kinetic model of the first and second stages of diffusion implies that the first stage of the process does not follow Fick’s first law. Therefore, it was assumed that the first stage of diffusion begins at $\beta = 0$ and continues until $\beta = 0.0766$, and from this point, the second step begins. According to formula (22), the second step continues until the value of $\frac{{C_{t}^{II} }}{{C_{\infty } }}$ does not reach 0.666. And, according to Table 2, this value corresponds to $\beta = 0.09943$.

3.2 The third step of diffusion of dye molecules in a cylindrical fibre

To clarify the nature of the third-stage diffusion process and to perform the necessary mathematical calculations [21], the author applied the model of the third-stage process presented in Fig. 3. When developing this model, it was considered that at the end of the second stage, when the dye molecules have penetrated the fibre in a given amount, $\left( \frac{2}{3} \right)C_{\infty }$, the third stage of diffusion begins.

Fig. 3

Cylinder without inverted cone

In Fig. 3, the volume of the inverted cone (brown), equal to $\frac{{C_{\infty } }}{3}$, is used to mathematically describe the rate of penetration of dye molecules into the fibre at the third stage of the dyeing process. A cylinder without an inverted cone, the volume of which is $\left( \frac{2}{3} \right)C_{\infty }$, shown in Fig. 3 is used for the first and second stage dyeing processes. It is assumed that at the third stage of the process of dye diffusion in the fibre, the diffusion intensity obeys the first Fick’s law. In this case, the concentration gradient of dye molecules both at the edges and in the centre of the fibre is determined by the following Eq. (48):

$${\text{d}}Q_{t}^{III} = D^{III} \left( {\frac{{\frac{1}{3}C_{\infty } - C_{t}^{III} }}{{r_{0} }}} \right)_{T} Sm_{0} {\text{d}}t^{III} ,$$

(48)

where ${\text{d}}Q_{III}$ is the mass of dye transported inside the fibres at the third diffusion stage during the time interval ${\text{d}}t^{III}$. $C_{t}^{III}$ is the concentration of dye molecules that have penetrated the fibres over an arbitrary period $t^{III}$, starting at the beginning of stage three $t^{III} = 0$. $D^{III}$ is the diffusion coefficient.$S = 2\pi r_{0}$—active fibre surface.

Given these variables, formula (48) can be rewritten as follows (49):

$${\text{d}}Q_{III} = D^{III} \frac{{\left( {\frac{1}{3}C_{\infty } - C_{t}^{III} } \right)}}{{r_{0} }}2\pi r_{0} m_{0} {\text{d}}t = 2\pi D^{III} \left( {\frac{1}{3}C_{\infty } - C_{t}^{III} } \right)m_{0} {\text{d}}t.$$

(49)

By applying the formula for the volume of a cone and placing the height of the given figure $C_{t}^{III}$, the following is true (50):

$$Q_{III} = \frac{1}{3}\pi r{}_{0}^{2} m_{0} C_{t}^{III} .$$

(50)

Differentiating (50) concerning $C_{t}^{III}$, we obtain the following Eq. (51):

$${\text{d}}Q_{III} = \frac{1}{3}\pi r{}_{0}^{2} m_{0} {\text{d}}C_{t}^{III} .$$

(51)

Comparing Eqs. (49) and (51) by ${\text{d}}Q_{III}$, we obtain the following Eqs. (52, 53):

$$\frac{1}{3}\pi r{}_{0}^{2} m_{0} {\text{d}}C_{t}^{III} = 2\pi D^{III} \left( {\frac{1}{3}C_{\infty } - C_{t}^{III} } \right)m_{0} {\text{d}}t,$$

(52)

$$\frac{{{\text{d}}C_{t}^{III} }}{{\frac{1}{3}C_{\infty } - C_{t}^{III} }} = 6\frac{D}{{r_{0}^{2} }}{\text{d}}t^{III} {\text{and}}\frac{{{\text{d}}C_{t}^{III} }}{{ - C_{t}^{III} + \frac{1}{3}C_{\infty } }} = 6\frac{D}{{r_{0}^{2} }}{\text{d}}t^{III}$$

(53)

Integrating right part (53), $C_{t}^{III}$ and left part $t$ we get the following (54):

$$- ln\left( { - C_{t}^{III} + \frac{1}{3}C_{\infty } } \right) = 6\frac{{D^{III} }}{{r_{0}^{2} }}t^{III} + c_{3} .$$

(54)

Assuming that the moment of the end of the second stage corresponds to the beginning of the third stage, we can calculate the value of the integration constant $c_{3}$. The beginning of the third stage of diffusion occurs at time $t^{III} = 0$. At that moment, $C_{t}^{III}$ is set to zero. Given that (54) we have the following (55):

$$c_{3} = \ln \frac{1}{3}C_{\infty } .$$

(55)

Adding $c_{3 }$ from (55) to Eq. (54), the following equations are true (56–60):

$$- {\text{ln}}\left( { - C_{t}^{III} + \frac{1}{3}C_{\infty } } \right) = - 6\frac{{D^{III} }}{{r_{0}^{2} }}t^{III} + {\text{ln}}\frac{1}{3}C_{\infty } ,$$

(56)

$$- {\text{ln}}\left( { - C_{t}^{III} + \frac{1}{3}C_{\infty } } \right) - {\text{ln}}\frac{1}{3}C_{\infty } = - 6\frac{{D^{III} }}{{r_{0}^{2} }}t^{III} ,$$

(57)

$${\text{ln}}\left( { - 3\frac{{C_{t}^{III} }}{{C_{\infty } }} + 1} \right) = - 6\frac{{D^{III} }}{{r_{0}^{2} }}t^{III} ,$$

(58)

$$3\frac{{C_{t}^{III} }}{{C_{\infty } }} = {\text{exp}} - \left( {6\frac{{D^{III} }}{{r_{0}^{2} }}t^{III} } \right),$$

(59)

$$\frac{{C_{t}^{III} }}{{C_{\infty } }} = \frac{1}{3}\left[ {1 - \exp - \left( {6\beta } \right)} \right].$$

(60)

Formula (60) is a kinetic equation for the dependence of the relative concentration of dye molecules penetrating the fibre in the third stage on the $\beta^{III} = \frac{{D^{III} }}{{r_{0}^{2} }}t^{III}$. To review the information and compare the values from dependency (60), Table 3 is compiled.

Table 3 Dependence $\frac{{C_{t}^{III} }}{{C_{\infty } }}$ of the dimensionless penetration time $\beta^{III}$ of diffusion process.

According to this table, the third stage of diffusion continues until the $\beta^{III}$ does not reach a value of about 0.8. The continuation of the third phase is approximately 35 times that of the second phase and eight times that of the first and second phases combined. Based on Tables 2–3, following the kinetic Eqs. (57, 58) of the diffusion of dye molecules into cylindrical fibres of radius $r_{0}$, plotted a hypothetical graph of the dependence of the relative concentration of penetrating dye molecules on the dimensionless penetration time $\beta = \frac{{\left( {Dt} \right)}}{{r_{0}^{2} }},$ in an isothermal process (Fig. 4).

Fig. 4

The hypothetical plot of the dependence of the relative concentration of penetrating dye molecules $\frac{{C_{t} }}{{C_{\infty } }}$ penetration time $\beta = \frac{{\left( {Dt} \right)}}{{r_{0}^{2} }},$ in an isothermal process

As can be seen from Fig. 4, the first stage is significant, i.e. it lasts until $\beta_{{{\text{finish}}}}^{I} = 0.0766$, starting from $\beta_{{{\text{initial}}}}^{I} = 0$. Comparing this theoretical result with the experimental result, it can be said that up to 57.88% of fisetin molecules accumulate on the fibre surface at the first stage. This result indicates the uniqueness of the partially crystalline supramolecular structure of fibroin microfibrils and is confirmed in Qiao et al. [22]. The second stage of diffusion begins with $\beta_{{{\text{initial}}}}^{II} = 0.0766$ and continues to $\beta_{{{\text{finish}}}}^{II} = 0.0994$. At this stage, 8.78% of fisetin molecules are transferred into fibroin fibres. The remaining 33.34% of dye molecules are transferred to the adsorbent in the third stage of the diffusion process. Calculations based on experimental and hypothetical diffusion kinetics plots show that the diffusion coefficients depend on the selected process step.

In conclusion, applying the kinetic model of diffusion, it should be noted that we have carried out calculations for isothermal processes. Thus, the results are acceptable for all processes of isothermal diffusion of monomolecular dyes in cylindrical fibres of radius $r_{0}$. In other words, it is possible to quickly determine the diffusion coefficient $D$ by measuring and plotting the experimental curve—as the primary data source—and using the hypothetical relationship curve shown in Fig. 4 as the calibration curve.

3.3 Suggestion a and its proofing

To determine the diffusion coefficient $D^{I}$ fisetin molecules, accumulated on active fibre surface in the first stage, we will use start and end points of interval $\Delta \beta^{I} = 0.0766$. As can be seen, absorption starts from $\beta_{{{\text{initial}}}}^{I} = 0$ and, following the hypothesis ends at $\beta_{{{\text{final}}}}^{I} = 0.0766$. At $\Delta \beta^{I}$ interval $\frac{{C_{t} }}{{C_{\infty } }}$ it becomes a value, equal to 0.5788. According to experimental curves, this process starts at $t_{{{\text{initial}}}}^{I} = 0$ and ends at $t_{{{\text{final}}}}^{I} = 5.95$ min, which means that during the whole period of the first stage, 57.88% of the dye mass is adsorbed on the fibre surface. Using these data and formulae (30 for $\beta$), we determine the diffusion coefficient $D^{I}$ of fisetin molecules on fibroin fibres at the first stage (61):

$$D^{I} = \frac{{\Delta \beta^{I} \cdot r_{0}^{2} }}{{\Delta t^{I} }},$$

(61)

where $r_{0}$—fibre radius; $\Delta \beta^{I}$– dimensionless time interval; $\Delta t^{I}$—time interval elapsed during the first stages of diffusion of dye molecules to the cylindrical sample.

The fibres in question are those of the mulberry silkworm Bombyx mori, whose average fibre diameter, free from sericin, wax and fatty substances, according to Rantano et al. [23] and Jiang et al. [24], is $\left( {12 \pm 2} \right)\mu {\text{m}}$ (i.e. $r_{0} = 7\mu {\text{m}}$). Let us set $D^{I}$ using these parameters (62):

$$D^{I} = \frac{{0.0766 \cdot \left( {7 \cdot 10^{ - 6} {\text{m}}} \right)^{2} }}{{5.95{\text{min}}}} = 6.31 \cdot 10^{ - 13} \frac{{{\text{m}}^{2} }}{{{\text{min}}}} = 1.05 \cdot 10^{ - 10} \frac{{{\text{cm}}^{2} }}{s}.$$

(62)

3.4 Suggestion b and its proofing

The second stage of the diffusion process determination of diffusion coefficient $D^{II}$ of fisetin molecules in fibroin fibres is based on the following principle: the second stage of diffusion is initiated by $\beta_{{{\text{initial}}}}^{II} = 0.0766$ and, according to the theory, ends at the value $\beta_{{{\text{final}}}}^{II} = 0.09943$. That is, this value $\beta$—of the interval is determined by the weight, $\frac{{C_{t} }}{{C_{\infty } }}$, equal to 0.666—corresponding to the end of the second phase (47): $\Delta \beta^{II} = 0.0229$. According to the experimental curves, this process starts at $t_{{{\text{initial}}}}^{II} = 5.95 min$ and ends at $t_{{{\text{finish}}}}^{II} = 7.12 {\text{min}}$. Continuation of the second phase $\Delta t^{II} = 1.17 min$, and during this interval 8.79% of the dye mass was transferred to the fibre.

Taking all these considerations into account, it is possible to determine $D^{II}$ using these parameters (63):

$$D^{II} = \frac{{\Delta \beta^{II} \cdot r_{0}^{2} }}{{\Delta t^{II} }} = \frac{{0.02283 \cdot \left( {7 \cdot 10^{ - 6} {\text{m}}} \right)^{2} }}{{1.17{\text{min}}}} = 1.59 \cdot 10^{ - 10} \frac{{{\text{cm}}^{2} }}{s}.$$

(63)

3.5 Proofing of suggestion c

According to the experimental isotherms, equilibrium colouration-saturation starts approximately at $\Delta t \approx 30 {\text{min}}$ after the beginning of the experiment. In the hypothetical dependence, this process corresponds to the value of the interval $\Delta \beta = 0.90712$. According to the hypothetical dependence—interval $\beta$ of diffusion at the third stage corresponds to the value of $\Delta \beta^{III} = 0.79792$. Determining the diffusion time for the third stage experimentally, we obtain $\Delta t^{III} = 22.88 {\text{min}}$. Now let us try to determine the diffusion coefficient mathematically. $D^{III}$ whereby according to the formula (30):

$$D^{III} = \frac{{\Delta \beta^{III} \cdot r_{0}^{2} }}{{\Delta t^{III} }} = \frac{{0.8 \cdot \left( {7 \cdot 10^{ - 6} {\text{m}}} \right)^{2} }}{{22.88{\text{min}}}} = 2.85 \cdot 10^{ - 10} \frac{{{\text{cm}}^{2} }}{s}.$$

(64)

Comparison of diffusion coefficient values $D^{I}$, $D^{II}$ and $D^{III}$ represented in the different stages indicates the change in the diffusion coefficient at each stage of the dyeing process (Fig. 5).

Fig. 5

Percentage of fisetin molecules penetrating fibroin at each stage of the staining process and the diffusion coefficient corresponding to each stage. Note: stage III—$D^{III} = 2.85 \cdot 10^{ - 10} \frac{{{\text{cm}}^{2} }}{s}$; stage II—$D^{II} = 1.59 \cdot 10^{ - 10} \frac{{{\text{cm}}^{2} }}{s}$; stage I—$D^{III} = 1.05 \cdot 10^{ - 10} \frac{{{\text{cm}}^{2} }}{s}$

Since the amount of dye penetrating the fibres is different at each stage of the process, it is possible to calculate the average values of the diffusion coefficient $\overline{D}$ fisetin molecules in natural silk fibres from the corresponding relative concentration values (65):

$$\overline{D} = \frac{{\left( {D^{I} \cdot 57.88\% + D^{II} \cdot 2.79\% + D^{III} \cdot 33.33\left( 4 \right)\% } \right)}}{100\% } = 1.6 \cdot 10^{ - 10} \frac{{{\text{cm}}^{2} }}{s}.$$

(65)

Equations (58–65) are semi-equal and involve complex relationships and approximations.

To calculate the mobility of fisetin molecules in a medium consisting of fibroin and water, we apply a more general equation that relates the mobility, diffusion and temperature of a given medium (66):

$$\overline{D} = \mu kT,$$

(66)

where $\overline{D}$ represents the diffusion coefficient, and $\overline{\mu }$—“mobility”, which means the ratio of the finite drift velocity and particle $\mathop \vartheta \limits^{{}}_{{{\text{drift}}}}$, to the applied strength $\vec{F},$ thus $\overline{\mu } = \frac{{\overline{\theta }_{{{\text{drif}}t}} }}{F}$–; $k -$ Boltzmann constant; $T$– absolute temperature.

Therefore (67):

$$\overline{\mu } = \frac{{\overline{D}}}{k \cdot T} = \frac{{1.6 \cdot 10^{ - 10} }}{{1.38 \cdot 10^{ - 23} \cdot 373}}\frac{{{\text{cm}}^{2} }}{s} \cdot \frac{K}{C} \cdot \frac{1}{K} = 3.11 \cdot 10^{6} \frac{{\text{m}}}{N \cdot s}.$$

(67)

For this system, in which dye and fibre interact, the relative characteristics of diffusion and adsorption play an important role in determining the rate of colour formation. The slow diffusion of fisetin dye molecules in natural silk fibres and the rapid sorption indicate the similarity between fisetin dye and fibroin fibre protein. This similarity between the dye and fibres is due to a variety of types of interactions, including hydrogen bonds, dipole–dipole interactions and van der Waals forces [25].

3.6 Suggestion d and its proofing

An auxiliary model was used to analyse the fibre dyeing process in the first and second stages. Assume that the material to be dyed is a plate with parallel surfaces and a certain thickness $2b$ (Fig. 6).

Fig. 6

Adsorbent in sheet form

To determine the part of the model volume occupied by the penetrated dye molecules, it is sufficient to subtract the prism volume from the total volume of the parallelepiped (68):

$$V^{{I\;{\text{and}}\;II}} = V_{{{\text{parallelepiped}}}} - V_{{{\text{prism}}}} = 2bhl - \left[ {\frac{{\left( {2b + 2\left( {b - x} \right)} \right)}}{2}} \right] \cdot hl = xhl.$$

(68)

When the dye molecules reach the centre of the parallelepiped plate $\left( {x = b} \right)$, $V^{I\& II} = bhl$, i.e. at the first and second diffusion stages, the dye molecules occupy half of the leaf space. By comparing this experimental curve with a hypothetical curve, the diffusion coefficient D is determined. In this case, the value of $\beta_{0.5} = 0.063$ will show the real-time value $t_{{{0}{\text{.5}}}}^{{{\text{real}}}}$, corresponding to half of the relative concentration value $\frac{{C_{t} }}{{C_{\infty } }} = 0.5$ (69):

$$D_{{{\text{sheet}}}} = \frac{{\beta_{0.5} \cdot b^{2} }}{{t_{{{0}{\text{.5}}}}^{{{\text{real}}}} }}.$$

(69)

Thus, the main challenge in determining the diffusion coefficient is to measure the change in the relative concentration of dye molecules in real time and create an experimental relationship.

3.7 Comparative analysis, long-term effects and scalability of dyeing processes

The long-term effects of the dyeing processes on textile materials were considered to assess the durability and sustainability of the techniques proposed. Accelerated ageing tests were conducted to simulate long-term use, and the results showed that textiles dyed with natural dyes retained their colour and structural integrity better over time compared to those dyed with synthetic dyes. Specifically, after 100 h of UV exposure, textiles dyed with fisetin retained 90% of their original colour intensity, while those dyed with Acid Red 1 retained 85%. The dye uptake for natural fisetin was measured at 85%, while a commonly used synthetic dye, Acid Red 1, showed a dye uptake of 80%. Additionally, the colour fastness tests indicated that the natural dye had a fastness rating of 4–5, whereas the synthetic dye had a rating of 4. Tensile strength tests indicated that the natural dye retained 95% of the original strength compared to 90% for the synthetic dye.

Scaling up the proposed dyeing processes was examined to identify potential challenges for industrial application. Key factors such as scalability of dye extraction, consistency in dyeing large batches and environmental impact were evaluated. The extraction process for natural dyes was optimised to ensure a consistent yield that can be scaled to industrial levels. Furthermore, pilot-scale trials were conducted to confirm that the dyeing process could be effectively scaled up without compromising dye uptake or colour consistency. Environmental assessments indicated that the natural dyeing process produced significantly less wastewater and required lower energy input compared to synthetic dyeing processes, making it a more sustainable option for large-scale operations. These considerations ensure that the proposed dyeing methods are not only effective on a laboratory scale but are also viable for industrial applications, promoting sustainability and efficiency in the textile industry.

3.8 Potential drawbacks and environmental impact of natural dyes in the textile industry

The potential drawbacks of using natural dyes in the textile industry must be carefully considered along with their benefits. One significant drawback is the variability in colour and consistency that can occur due to differences in the source material. Unlike synthetic dyes, which can be produced with precise chemical formulations, natural dyes depend on the plant or animal source, which can vary with growing conditions, harvest times and processing methods. This variability can lead to inconsistencies in the final product, making it challenging to achieve uniformity in colour across large batches of textiles.

Another concern is the environmental impact of extracting natural dyes. While natural dyes are often promoted as an eco-friendly alternative to synthetic dyes, the extraction process itself can have significant environmental repercussions. Large-scale cultivation of dye-producing plants can lead to deforestation, loss of biodiversity and soil degradation. Additionally, the extraction and processing of these dyes often require large amounts of water and energy, which can contribute to environmental pollution if not managed sustainably. For instance, the use of solvents in the extraction process can result in chemical waste, which needs to be treated properly to prevent contamination of water sources. The safety and health implications of using natural dyes are also critical considerations. While natural dyes are generally perceived as safer for both workers and consumers compared to synthetic dyes, some natural dyes can cause allergic reactions or skin irritation. Additionally, the handling and processing of natural dyes may expose workers to various biological and chemical agents, necessitating proper safety protocols and protective equipment to ensure their well-being. It is essential to conduct thorough safety assessments and implement stringent health standards to mitigate any potential risks associated with the use of natural dyes in the textile industry.

To address these challenges and reduce the environmental impact of the textile industry, several alternative solutions can be explored. One promising approach is the development of sustainable agriculture practices for cultivating dye-producing plants. This includes organic farming techniques that avoid the use of harmful pesticides and fertilisers, as well as crop rotation and polyculture to maintain soil health and biodiversity. Additionally, advances in biotechnology could lead to the development of genetically modified plants that produce higher yields of dye compounds, reducing the land and resources needed for dye production. Another solution is the implementation of closed-loop systems in the dye extraction and dyeing processes. These systems aim to recycle water and solvents used in the extraction process, minimising waste and reducing the overall environmental footprint. Innovations in dyeing technology, such as waterless dyeing methods and the use of supercritical CO₂, also offer potential pathways to significantly decrease water and energy consumption in the dyeing process. Furthermore, the adoption of digital printing technologies in textile production can reduce the need for large volumes of dyes and water. Digital printing allows for precise application of dye to the fabric, reducing waste and enabling greater design flexibility. This technology can also facilitate the use of natural dyes by enabling small-scale production runs and customisation, aligning with the variability and unique characteristics of natural dye sources.

In conclusion, while natural dyes offer a sustainable alternative to synthetic dyes, their use in the textile industry comes with certain drawbacks and environmental impacts that need to be carefully managed. By exploring sustainable agricultural practices, closed-loop systems and advanced dyeing technologies, the industry can mitigate these challenges and move towards more environmentally friendly and safe dyeing processes. Ensuring the well-being of workers and consumers through rigorous safety standards and innovative solutions will be crucial in promoting the widespread adoption of natural dyes in textile production.

4 Discussion

The presented study focuses on the specific dye fisetin and its interaction with fibroin fibre. This is important because different dyes may have different diffusion mechanisms, and understanding these features is key to developing optimal dyeing methods. An interesting result is the increase in the diffusion coefficient in the third stage, despite the slowdown of mass transfer. This leads to the question of what processes and factors may slow down mass transfer despite the increase in diffusion rate. Burada et al. [26] concluded that for particles undergoing displacement diffusion in static suspension media confined in confining geometries, transport exhibits intriguing features such as a decrease in nonlinear mobility with increasing temperature or a broad excess peak of effective diffusion above the free diffusion limit. These paradoxical phenomena can be explained in terms of entropic contributions due to confined dynamics in phase space.

According to Lee et al. [27], the study of the antioxidant properties and free radical scavenging ability of biomaterials is of great importance in the context of tissue engineering and regenerative medicine. In this context, epigallocatechin gallate, known as an effective antioxidant, plays a key role. This study aims to evaluate the potential of biomaterials coated with epigallocatechin gallate for tissue engineering applications. One important aspect of the study is to investigate the antioxidant properties of epigallocatechin gallate and its effect on biomaterials. Antioxidants can help protect tissues and biomaterials from oxidative stress and damage, which is critical for successful tissue regeneration. All of this contributes to the development of biomaterials with higher stability and the ability to maintain a healthy environment for cells during the tissue engineering process. This study not only expands the knowledge of antioxidant interactions with biomaterials but may also lead to the development of more efficient and stable materials for regenerative medicine and tissue engineering applications.

Referring to the definition of Pecorini et al. [28], flavonoids are naturally occurring compounds that often have potential therapeutic properties such as antioxidant and anti-inflammatory effects. However, their effective use in medical applications is often limited by the need for controlled and long-term release. Polymeric systems represent a promising method to achieve this goal. They can be engineered to release flavonoids in a controlled manner over long periods, allowing the levels of active ingredients in the body to be maintained within a therapeutic range. This approach is of great importance for the treatment of chronic diseases, such as cardiovascular disease or cancer, where continuous and long-term flavonoid therapy may be necessary. These findings are consistent with the theses presented in this study. Polymeric systems can also protect flavonoids from degradation in the gastric environment and ensure their delivery to the right place in the body. This can significantly improve the bioavailability and efficacy of flavonoids as medicines. Thus, research into polymeric systems for the controlled release of flavonoids has the potential to change the paradigm in the field of pharmacology and open new avenues for the treatment of various diseases.

Velázquez-Contreras et al. [29] determined that studies related to the use of cyclodextrins in polymer-based active food packaging reflect important trends in food technology and sustainable solutions. Cyclodextrins are ring-shaped molecules capable of forming inclusions with various aromatic and functional compounds, making them ideal for capturing and preserving flavour and nutrients in food products. This approach is of great importance in the grocery industry, where preserving product quality throughout the shelf life is crucial. Beyond this, however, the use of cyclodextrins also fits into the trend towards more sustainable and biodegradable food packaging. Polymeric materials incorporating cyclodextrin can be more sustainable and environmentally friendly. This is in line with current requirements for sustainable and nontoxic packaging technologies, which is relevant in light of the growing interest in environmental care and waste reduction. By analysing the results obtained and the conclusions drawn, research in this area not only contributes to improving packaging quality and food preservation but also promotes the development of more sustainable and environmentally friendly packaging technologies, which is important for the food industry and society as a whole.

B. Balachandran and Sabumon [30] determined that the use of natural dyes and bio-etching agents in the textile industry represents an important step towards cleaner and more environmentally sustainable production of textile materials. The textile industry has historically often used synthetic dyes and chemical mordants, which can cause serious environmental and human health problems. However, natural dyes derived from plants, minerals, and animals are more environmentally friendly and biodegradable. Their use reduces the negative impact of the textile industry on nature. Moreover, they often have antimicrobial and antioxidant properties, which can be useful for the production of functional textile materials. Bio-etching agents, also known as enzymes, are an alternative to chemical etchants and can be a gentler and more effective means of treating textile fibres. They can be used to bleach, clean, and treat fabrics while reducing harmful emissions and waste. The use of natural dyes and bio-etching agents in the textile industry helps to create cleaner production, reduce pollution and create more sustainable production methods. This is of great importance not only for the environment but also for consumers, who are increasingly attentive to the origin and environmental performance of textile products.

As noted by Odero et al. [31], the use of natural dyes, especially those derived from plants, can have several advantages. They can be more environmentally friendly than synthetic dyes and contribute to the reduction of pollutant emissions. The heartwood of the Prosopis juliflora tree contains bioactive substances that may have colouring properties. Exploring their potential for dyeing cotton fabric opens up new possibilities for utilising nature’s resources in the textile industry. In addition, such natural dyes can give textiles a special character and unique appearance, which may interest consumers looking for exclusive and environmentally friendly products. This research contributes to the diversity of dyeing methods and highlights the importance of developing sustainable and environmentally friendly technologies in the textile industry.

Zasada-Kłodzińska et al. [32] proved that research aimed at analysing natural dyes on historical objects using high-performance liquid chromatography and electromigration techniques is of great importance for understanding and preserving historical man-made objects and works of art. Many historical dyes have been derived from plants, minerals and animals, and their analysis can unlock the mysteries of art history and the technology of past eras. High-performance liquid chromatography and electromigration techniques provide accurate and reliable analyses of the components of natural dyes, allowing researchers to determine their composition and origin. This information can be very useful for reconstructing the colouring methods used in historical works of art. Moreover, the analysis of natural dyes on historical objects contributes to the conservation and restoration of man-made objects. Understanding which dyes were used allows for a proper approach to their conservation and restoration. This increases our respect for historical heritage and helps to preserve it for future generations.

Chen et al. [33] utilised an all-atom kinetic Monte Carlo model to simulate the chemical vapour deposition growth of graphene on a Cu substrate. Their study emphasised the importance of accurately capturing atomistic events, such as the deposition and diffusion of carbon species, which significantly affect the morphology of the grown graphene. Similarly, our study delves into the detailed kinetics of dye diffusion, focussing on the adsorption, diffusion and uniform distribution of fisetin molecules in fibroin fibres. Both studies underscore the necessity of including detailed kinetic events to predict and optimise process outcomes. However, while Chen et al. focussed on the atomic-level processes in a solid-state reaction, our research examines molecular interactions in a liquid-phase dyeing process. The introduction of specific kinetic stages in our model aligns with the inclusion of edge attachment and diffusion events in Chen's model, demonstrating a parallel in the complexity and depth of analysis required to understand and control these processes.

Al-Raeei [34] investigated the specific bond volume using Morse potential and its dependence on various factors, including temperature and molar volume. Their findings showed that the specific bond volume decreases with decreasing temperature and varies with the depth of the Morse potential well. This focus on temperature dependence resonates with our study's consideration of temperature as a critical factor influencing dye diffusion rates. Both studies highlight the importance of temperature control in determining the properties and behaviour of materials. In our research, maintaining a consistent temperature was crucial to accurately measuring the diffusion coefficients and understanding the kinetics of dyeing. Al-Raeei's approach to modelling and quantifying molecular interactions through specific bond volume complements our method of using spectrophotometric measurements and kinetic equations to quantify dye concentration and diffusion stages.

Qian et al. [35] reconstructed thermodynamic equations for reaction processes, applying them to differential scanning calorimetry and differential thermal analysis. Their study introduced a new thermodynamic equation that better reflects the dynamic nature of reaction processes by incorporating multiple complex factors, such as species, mass and heat capacity. This approach is analogous to our method of developing a kinetic model that accounts for the stages of dye diffusion and the associated concentration gradients. Both studies aim to enhance the accuracy and reliability of predictive models by incorporating a comprehensive set of influencing factors. While Qian et al. focussed on thermodynamic properties and reaction kinetics in chemical processes, our study concentrated on the kinetic aspects of dye diffusion in textile fibres. The common thread is the use of detailed modelling to improve process understanding and predictability, ultimately leading to more efficient and controlled outcomes.

Thus, the shift towards natural dyes and bio-etching agents in textile production underscores a pivotal step towards environmentally friendly practices [36]. Insights from molecular analysis techniques in previously dye studies further highlight the relevance of detailed kinetic modelling, akin to our approach in delineating dye diffusion stages. Such integrated perspectives underscore the complexity and potential for optimising industrial processes through advanced kinetic modelling and interdisciplinary insights.

5 Limitations

One limitation of this study is the difficulty in accurately measuring dye concentration over time, which may have introduced some uncertainty into the results. Additionally, there may have been errors in estimating diffusion coefficients using mathematical modelling. Although based on established theories, any assumptions and simplifications could affect accuracy. Despite these limitations, the study offers valuable insights into dye diffusion kinetics and provides a solid foundation for further research to optimise dyeing processes and promote sustainability in the textile industry.

6 Implications

The economic feasibility of implementing the proposed techniques for natural dyeing in the textile industry is a critical consideration. While natural dyes can be more expensive than synthetic dyes due to the cost of raw materials and extraction processes, the long-term savings from reduced environmental impact and potential regulatory benefits can offset these initial costs. Additionally, as the market for sustainable and eco-friendly products grows, the demand for textiles dyed with natural dyes is likely to increase, potentially leading to economies of scale and lower costs over time. Investments in efficient extraction and dyeing technologies can further improve the economic viability of natural dyes. The findings of this study have significant implications for real-world dyeing processes. By understanding the three-stage kinetic model of dye diffusion, manufacturers can optimise dyeing methods to achieve more consistent and high-quality results. This model allows for better control over the dyeing process, reducing waste and improving the efficiency of dye usage. Implementing these optimised techniques can lead to higher productivity and lower costs, making natural dyes a more attractive option for textile producers.

Culturally and socially, the use of natural dyes can have profound implications. Natural dyes are often associated with traditional and artisanal textile practices, which can enhance the cultural value of textile products. By integrating natural dyes into modern manufacturing, the textile industry can help preserve cultural heritage and support local economies, especially in regions where dye plants are cultivated. This cultural resonance can appeal to consumers seeking authentic and ethically produced goods, thereby enhancing the marketability of products dyed with natural dyes.

The study's findings also have implications for the competitiveness of the textile industry. As consumers become more environmentally conscious, companies that adopt sustainable practices, including the use of natural dyes, can differentiate themselves in the marketplace. This competitive edge can lead to increased market share and customer loyalty. Additionally, compliance with emerging environmental regulations and sustainability standards can further enhance a company's reputation and competitiveness.

Regulatory requirements for using natural dyes in textile production are another important consideration. The use of natural dyes must comply with safety and environmental regulations, which may vary by region. Ensuring that natural dyes meet these regulatory standards is crucial for their widespread adoption. This includes verifying the absence of harmful substances and ensuring that the extraction and dyeing processes do not negatively impact workers or the environment. By adhering to these regulations, textile manufacturers can avoid potential legal issues and enhance the overall sustainability of their production processes.

In conclusion, the implications of this study extend across economic, cultural, social and regulatory dimensions. By optimising the use of natural dyes through a better understanding of dye diffusion kinetics, the textile industry can achieve economic benefits, cultural preservation, competitive advantages and regulatory compliance. These factors collectively support the broader adoption of natural dyes, contributing to a more sustainable and responsible textile industry.

7 Conclusions

This study includes a comparison between natural and synthetic dyes to determine their relative effectiveness in textile applications. Experiments were conducted using both types of dyes under identical conditions to evaluate their performance. The results show that natural dyes, such as fisetin, exhibit superior dye uptake and colour fastness compared to synthetic dyes. Also, textiles dyed with natural dyes retained their colour and structural integrity better over time compared to those dyed with synthetic dyes. Additionally, tensile strength tests indicated that the natural dye did not significantly degrade the fibre strength. These findings highlight the potential of natural dyes as effective alternatives to synthetic dyes in textile applications.