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Section 7. Physics
Shukurlu Yusif, Sheki Regional Scientific Center of the National Academy of Sciences of the Republic of Azerbaijan Sheki, Azerbaijan E-mail: [email protected]
THE THIRD STAGE OF THE DIFFUSION PROCESS OF FISETIN MOLECULE IN THE FIBROIN FIBER
Abstract: The previous article stated that the diffusion of the dye continues until it reaches an equilibrium concentration in the entire volume of the fiber. This period was mentally divided into three stages: 1) adsorption of fisetin molecules on the surface of fibroin fibers; 2) moment when fisetin molecules reach the center of the fibroin fiber. Since the first stage occurs almost instantaneously, it is almost impossible to separate this stage from the second stage during which the actual dyeing occurs. We combined the first and second stages of diffusion and devoted a previous article to them. The third stage begins after the completion of the second and continues until the equilibrium concentration is restored in the entire volume of the fiber. This article is focused on the third stage of diffusion of flavonoid fisetin molecules in the fibroin fiber. This article also uses a three-dimensional physical model of the diffusion of dyes in the fiber. We established mathematical relations that describe the kinetics of dye diffusion at the third stage of the dyeing process.
Keywords: dye solution temperature; electrolyte concentration in the solution; diffusion kinetic parameters; physical model of distribution; kinetic characteristic of diffusion.
Results and discussion (continuation of[1]) the dye diffusion intensity in the fiber at the third
By using the three-dimensional model and car- stage of the process is determined by the first Fick's
rying out simple mathematical calculations, it can be law, the difference in concentration potentials at the
shown that at the moment when the dye reaches the edges and in the center of the fiber, can be composed
center of the fiber, the concentration of penetrating as the following equation: dye molecules is equal to 2Cm /3 . The model uses 1 c - r111
3 M
figure with volumes equal to 2nr02 /3 (Fig. 6) to ex- dQiii = D Sm0dt, (21)
ro
press the concentration. As one can see during the i . ... rri . . i
r & where d.Qm is mass quantity or fiber transported over
third stage, the remaining volume with concentra- .. . a, j cj-ir
& ; & time dt, inside the fibers at the third stage oi dittu-
tion: C111 is equal to nrO /3, and in this volume the . ^iii ■ ... ri i i
t n 0 > sion; Ct is the concentration of dye molecules penmaximum concentration should be equal to one etrating during an arbitrary period of time t, that has third of the concentration of the dye at the fiber passed since the beginning ofthe third stage; D is the edges: CM, which is equal to Ca /3. Consequently, diffusion coefficient; S is the active area of the fiber.
stage ofthe diffusion process begins at tm = 0. At that moment Cf1 is equal to 0. Considering that (24):
Figure 6. Three-dimensional physical model of the dye diffusion in the fiber at the third stage of dyeing
In order to make sure that S, which is used in the formula (21), and determines area of the active fiber, is equal to S = 2nr0, we shall take note of three-dimensional model of fiber (Fig.6) In this model, the height of fibroin fiber shown as a transparent cylinder, equals to one. Taking this into account, the formula (21) can be written as: 1
c 3 =- ln3 C»•
By adding c3 - (the constant of integration) to (24), we obtain the following equation:
/
- ln
C - CI
1C - CL
V 3
dQni = D
3
-2nr0m0dt =
(21 a)
D
6-t =ln
= 2nD
1
m0dt.
-C - CL
Q W t
V 3
Figure 6 shows that the mass quantity of the transported dye at the third stage of diffusion can be expressed by the following formulas:
Qni = 3 nr2m0CfL. (22)
By differentiating (22) with respect to Cf1, the following equation is obtained:
or
= 6 D t - lni C „ r0 3
3C"
1
a - cl
Ct c „
1 - exp
-6 D1
V r0 J
(25)
(26)
dQiii = 3nr0Zm0dCL1
(22 a)
By comparing (21a) and (22a) with respect to dQin, the following eqaution is obtained:
(23)
dC
D
= 6—dt •
^c - cll 3 C" Ct
By integrating right-hand side of (23) to C11, and left-hand side to dt, we obtain the following:
' * (24)
- ln
1
D
= 6—t + c.
-C - CI
O TO t
V 3 . o
By accepting the end of the second stage and beginning of the third stage as check time, we determine the value of the integration constant - c3. The third
(26) is the kinetic equation for the concentration of dye molecule that penetrates the fiber at the third stage of the diffusion process.
The kinetic dependence of the relative concentration of fisetin molecules in the fibroin fiber at the third stage of the diffusion process, which is shown in (Figure 7), was constructed based on this equation.
The kinetic equation for the diffusion of dye molecules into fibers (26) at the third stage of the process is especially significant due to the fact that by applying these equations at a given temperature and concentration of the dye, the diffusion coefficient D can be determined. The results of our measurements were used to determine D .
As shown in Figure 1 (in previous article), molecules of fisetin dye that penetrate fibroin fibers introduced into the dye solution at boiling point (373 K) have the maximum concentration equal to
0
0
3
0
Cmax « 0.56g / kg . This concentration is established
in 30 minutes. The comparison shows that the concentration of penetrating dye molecules in the fiber in the second and third stages of diffusion is equal to
C11 :Cin = 2:1. Using this dependence and knowing
that C = C + C , we can construct the following
<0.37g / kg and Ca* « 0.19g / kg.
equations:
cL
Figure 7. Kinetic curve of the relative concentration of the fisetin dye molecules in fibroin fiber at the third stage of diffusion
We used 4g of fisetin to prepare an aqueous solution of the fisetin dye for each 100g of fibroin. Consequently: Cx =40g/kg.
To caculate average value of D, we use the ratios of the volumes of the second and third stages equal to 2:1, consequently, the ratios between the time of fiber dyeing at the second and third stages is also 2:1. Therefore, by inserting the values of the duration of
the second stage: tn = 20minutes and the duration ofthe third stage: tin = 10minutes, the cross-sectional area of the natural silk fiber is ~370 ^m2 and by accepting the shape of the fiber as a cylinder, we obtain the value for the radius of the cylinder
r0 = 10.8 -10-
m.
To calculate D by (25), we use the following values:
r2 A D = ^ln
6t
C
C - Cin
V w max y
118 -10"1 6 • 600
ln
40 ^m!=1.56 •10-1«mL=1.56•10-12m
V
40 - 0.19
By comparison, for a rough safety assessment, diffusion coefficients were measured for 13 radioactive elements in clay media. Authors have found that D values range from 5.0-10-nm2 / s for I and Tc (under oxidation conditions) to 5.0-10-14m2 / s for U, Np and Pu actinides (under reducing conditions) [2]. This shows that the fibroin fiber exhibits viscosity to fisetin molecules hundreds of times higher than clay media to radioactive elements.
s s s
We use Nernst-Einstein equation that established relationship between mobility, diffusion, and temperature of the medium to calculate the mobility of the fisetin dye molecule in the fibroin+water medium [3]: D = ukT,
where u is the mobility of molecules, k is the Boltzmann factor, T is the temperature of the medium. Consequently:
u = -
D
1.56-10-
m'
kT 1.38-10-23 • 373 s Consequently, the mobility of fisetin molecules in the fibroin+water medium at the temperature of 373 K is 3.03 • 104 m/(Ns).
Due to the fact that in dye-fiber system, the relativity properties of diffusion and sorption determine the color formation rate [3] and correspond to the slow diffusion of the fisetin dye molecules into natural silk fiber (d = 1,56-10-16m2 /s) andhigh-speed sorption (u = 3.03-104 m/(Ns) demonstrate affinity between the fisetin dye and the fibroin protein.
As kinetic and thermodynamic parameters, there is a very complex relationship between the diffusion rate and the affinity between the dye and the fiber [4]. Textile materials have a specific requirement for the fiber and dye diffusion and sorption processes must simultaneously be active during interaction. Color cannot be formed if any of those conditions is not met.
Electrostatic forces are undoubtedly affect the diffusion process. It is generally believed that a moderately high concentration of a medium high concentration of low-molecular electrolyte, such as
• — • — = 3.03-104 m / (N • s). J K v '
NaCl, will remove any such effects and it is confirmed by the fact that D at moderately high ionic forces, becomes virtually non-affected by the total charge [5]. However, the added electrolyte, apparently, will have minor effect if it (the added electrolyte) initially has the universally identical concentration [6].
Another important point is that, on the one hand, we are trying to create dyes with increased affinity for the fiber, since this provides high dyeing fastness to wet treatments, on the other hand, the increased dye affinity for the fiber reduces the diffusion rate, and, consequently, speed of dyeing process. This contradiction can be overcame in real conditions by building the technological process to ensure a decrease in the affinity of the dye at the time of the fiber entering the dye solution and to create conditions for the manifestation of this affinity after the diffusion is completed. Temperature changes, solvation of dye with auxiliary substances of hydro-philic solvents, etc. are used for these purposes.
References:
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2. Michael J. Stenhouse, Thierry Merceron, Edouard Scott de Martinville. Provision of diffusion coefficients for argillaceous media in support of preliminary safety assessment within the French HLW disposal programme // Journal of contaminant hydrology,- Vol. 21.- Issues 1-4. 1996.- P. 351-363. Tinker P. B., Nye P. H. Solute movement in the rhizosphere, 2nd edn. USA: Oxford University Press,-New York, 2000.- 370 p.
Мельников Б. Н., Виноградова Г. И. Применение красителей. Учеб. для вузов - М.: Химия, 1986.240 с.
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