Application of an equation derived from the solution of diffusion equation with constant surface concentration to the film-roll method

Abstract

In this study, an equation to obtain reliable diffusion coefficients for the sublimation diffusion of disperse dye in paste into polyethylene terephthalate film using the film-roll method was derived from the solution of the diffusion equation with the constant surface concentration. The constant surface concentration in the equation, where the total amount of dye passed through the first-layer surface is a linear function of the square root of time, was obtained by the steady-state straight line for a sufficiently long heat treatment time. The diffusion coefficients were obtained by the constant surface concentration and the slope of the linear regression line for the plot of the total amount against the square root of time. Their reliability was proved by the good linearity of linear regression for the Arrhenius plot. The activation energy for the diffusion coefficients obtained by this equation was similar to that obtained by another equation derived from the solution with the time-varying second-layer surface concentration. The diffusion coefficients for data containing $(0, 0)$ in the plot of the total amount against the square root of time were judged to be more reliable than those for data not containing $(0, 0)$ .

Keywords

Diffusion printing dyeing sorption color finishing

The film-roll method is very useful for studying the kinetic behaviors of various dyes on polymer substrates and has long been used to measure the diffusion coefficients because the diffusion behaviors can be precisely observed and the ratio of concentration between adjacent layers can be easily determined by the concentration–distance curve.^1–3 In a study on the kinetic behavior of vat dyes in cellulose, Sekido and Matsui⁴ proposed a method for determining the diffusion coefficient by obtaining the variable $ζ$ of the iterated complementary error function, $ierfc (i ζ)$ , from the ratio of the mean concentration between adjacent layers, ${\bar{C}}_{i + 1} / {\bar{C}}_{i}$ , where $ζ = ε / \sqrt{4 D t}$ , ${\bar{C}}_{i}$ is the mean concentration of the $i$ th layer, $ε$ is the thickness of layer, $t$ is the time, and $D$ is the diffusion coefficient.

In a study on the sublimation diffusion behaviors of disperse dyes for transfer printing, Jones and Leung⁵ proposed a technique for determining the diffusion coefficient by inserting a transfer-print paper and two films between the top and bottom hot plates, by constantly replacing the transfer-print paper at the top and the film at the bottom with fresh paper and film every 15 seconds, and by applying the equation $L = l^{2} / (6 D)$ where $L$ is the time lag, $l$ is the thickness of film, and $D$ is the diffusion coefficient. However, this method was judged to be inconvenient and less accurate, so a new method to obtain the diffusion coefficient for the sublimation diffusion was needed.

It was expected that the film-roll method could also be applied to sublimation diffusion, so the factors influencing the sublimation diffusion of disperse dye in paste into polyethylene terephthalate (PET) film were investigated using the film-roll method,⁶ and the diffusion coefficients were determined by obtaining the variable $ζ$ of $ierfc (i ζ)$ from ${\bar{C}}_{i + 1} / {\bar{C}}_{i}$ .^7,8 According to the Arrhenius plot, the common logarithm of the diffusion coefficient is the linear function of the reciprocal of absolute temperature, so the correlation coefficient $(r)$ of linear regression for the Arrhenius plot should be close to 1. However, the obtained $r$ was 0.96, which means that the linearity of linear regression is low, so the reliability of the diffusion coefficients was doubtful. Thus, it was necessary to find another equation for obtaining the diffusion coefficient in the sublimation diffusion using the film-roll method.

In the first attempt to obtain the diffusion coefficient for the sublimation diffusion of disperse dye in paste into PET film, $C (x, t) = C_{1} erfc (x / \sqrt{4 D t})$ was applied to the film-roll method, where $C (x, t)$ is the concentration at position $x$ for time $t$ and $C_{1}$ is the surface concentration. The mean diffusion coefficients were determined by averaging the diffusion coefficients obtained by $x / \sqrt{4 D t}$ from $C (x, t) / C_{1}$ at a specified time.⁹ The Arrhenius plot for the mean diffusion coefficients showed good linearity $(r = 0.9991)$ . However, the reliability of the mean diffusion coefficients was questionable because the standard deviation of the diffusion coefficients was large. Also, the criteria to determine the constant diffusion coefficient are ambiguous.

In the second attempt, on the premise that the steady-state condition was set up at a sufficiently long heat treatment time, the equation for obtaining the diffused total amount by the steady-state first-layer surface concentration and the steady-state diffusion length was derived from the solution of the diffusion equation in the form of a trigonometrical series.¹⁰ The linearity of the diffusion coefficients obtained from the steady-state straight lines was very good $(r = 0.9999)$ . However, they cannot be the constant diffusion coefficients at each temperature because they are just the diffusion coefficients for time in a steady state.

In the third attempt, the equation for obtaining the steady-state first-layer-passed total amounts passed through the second-layer surface was also derived from the solution of the diffusion equation in the form of a trigonometrical series. The diffusion coefficients were determined by the time lag obtained from the plot of the steady-state first-layer-passed total amounts against time, the steady-state first-layer surface concentration, and the steady-state second-layer surface concentration.¹¹ The linear regression line for the Arrhenius plot had good linearity $(r = 0.9961)$ . However, they also seem difficult to be the constant diffusion coefficients at each temperature because of the unusually large diffusion coefficient at 190°C and the relatively large activation energy.

In the fourth attempt, on the premise that the second-layer surface is the surface for the diffusion of the time-varying surface concentrations, the equation for obtaining the total amount passed through the second-layer surface was derived from the solution of the diffusion equation for the diffusion of the time-varying second-layer surface concentrations.¹² The reliability of the diffusion coefficients for the diffusion of the time-varying second-layer surface concentrations was identified by the Arrhenius plot $(r = 0.9914)$ .

In this study to find an equation for obtaining a more reliable diffusion coefficient for the sublimation diffusion of disperse dye in print paste into PET film using the roll method, the equation for obtaining the total amount passed through the first-layer surface was derived from the solution of the diffusion equation for a semi-infinite medium whose surface was maintained at a constant surface concentration. The constant surface concentration was obtained by the steady-state straight line at a sufficiently long heat treatment time. The reliability of the diffusion coefficients obtained by the constant surface concentrations and the slopes of the linear regression lines for the plots of the total amounts passed through the first-layer surface with respect to the square roots of times was proved by Arrhenius plots and by comparison with the diffusion coefficients for the diffusions of the time-varying second-layer surface concentrations.

Materials and method

Materials

Biaxially oriented PET film with a thickness of 40 μm was supplied by SKC Co. Ltd (Suwon-si, Gyeonggi-do, Rep. of Korea). Sumikaron Bordeaux SE-BL (C. I. Disperse Violet 26, Sumitomo Chemical Co. Ltd) was used for disperse dye without purification. Sodium alginate (Duksan Chemicals, Inc.) of extra pure grade was used.

Preparation of paste and film roll

The paste was prepared by mixing 35 g of sodium alginate, 300 g of dye, and 665 g of water. The dye concentration was determined as the concentration satisfying the infinite dye-bath condition through preliminary experiments. A schematic diagram illustrating the film-roll preparation process is shown in Figure 1. The thickness of the film and the height of the film layers were enlarged at a ratio of 25:1 to the size of the glass rod. The PET film of 0.004 cm thickness and 58 cm length was wound tightly around a glass rod of 2 cm diameter (a). Both edges of the film roll were wound with aluminum tape of 0.5 cm width to make barriers of 0.018 cm height. The paste containing dye was primarily coated on the surface of the film roll between both barriers to be about 0.01 cm thick and was dried in a dryer at 100°C for 5 minutes (b). The paste was finally coated on the surface of the primarily coated paste to be 0.018 cm thick and was dried at 100°C for 5 minutes. The surface of the paste-coated film roll was wrapped with aluminum tape to prevent the paste from falling off the film surface and to prevent the dyes in the paste from sublimating into the air (c). A round glass rod was pressed and both barrier parts of the glass rod were tied tightly with cotton yarn to prevent dye penetration for dividing the film layers (d).^9–12

Figure 1.

Schematic diagram illustrating the overall film-roll preparation process, including the paste coating. PET: polyethylene terephthalate.

Heat treatment

The heat treatment of the film roll was performed in a laboratory curing chamber at 170°C, 180°C, and 190°C for various times. The heat-treated film roll was washed with water after removing the coated paste, unrolled, and cut into the section pieces.

Determination of mean dye concentration

To extract the dye in the film, each section piece was immersed individually in a test tube containing chlorobenzene of 20 mL and heated at 132°C until the dye was completely extracted.¹³ The optical density at the maximum absorption wavelength of the dye was measured using a spectrophotometer (Agilent Cary 8454) and the mean dye concentration $(mg \cdot {cm}^{- 3})$ of each layer was obtained from the calibration curve.

Results and discussion

Diffusion equation with constant surface concentration

When the diffusion coefficient is constant, the diffusion equation for the diffusion in one dimension is given by

\frac{\partial C (x, t)}{\partial t} - D \frac{\partial^{2} C (x, t)}{\partial x^{2}} = C_{t} (x, t) - D C_{x x} (x, t) = 0 (1)

where

x

is the distance from the surface,

t

is the time,

D

is the diffusion coefficient, and

C (x, t)

is the concentration at position

x

at time

t

. The solution of the diffusion equation for a semi-infinite medium whose surface is maintained at a constant concentration is obtained by the dilation property of the diffusion equation. Suppose Equation (1) satisfies the following initial and boundary conditions

C (x, 0) = 0 for t = 0 and 0 < x < \infty

(2)

C (0, t) = C_{1} for x = 0 and t > 0

(3)

where

C_{1}

is a constant surface concentration. If

a

is a positive constant and

C (x, t)

is a solution of the diffusion equation, then the dilated function

E (x, t) = C (\sqrt{a} x, a t)

can be also a solution of the diffusion equation.¹⁴ This can be proved as follows

\frac{\partial E (x, t)}{\partial t} = \frac{\partial C (\sqrt{a} x, a t)}{\partial t} = a C_{t} (\sqrt{a} x, a t)

(4)

\frac{\partial^{2} E (x, t)}{\partial x^{2}} = \frac{\partial}{\partial x} [\frac{\partial C (\sqrt{a} x, a t)}{\partial x}] = \frac{\partial}{\partial x} [\sqrt{a} C_{x} (\sqrt{a} x, a t)] = a C_{x x} (\sqrt{a} x, a t)

(5)

\frac{\partial E (x, t)}{\partial t} - D \frac{\partial^{2} E (x, t)}{\partial x^{2}} = a [C_{t} (\sqrt{a} x, a t) - D C_{x x} (\sqrt{a} x, a t)] = 0 (6)

(6)

By this dilation property, a solution $C (x, t) = f (x / \sqrt{t})$ is unaffected by the dilation $x \to \sqrt{a} x$ and $t \to a t$ , because the quantity $x / \sqrt{t}$ is unaffected by the dilation due to $\sqrt{a} x / \sqrt{a t} = x / \sqrt{t}$ . Therefore, the solution $C (x, t) = f (x / \sqrt{4 D t})$ is also unaffected by the dilation.

In order to obtain a special solution of the form $C (x, t) = f (x / \sqrt{4 D t})$ , let $f (x / \sqrt{4 D t})$ be a function of one variable $ζ$ , that is, $C (x, t) = f (ζ)$ . Then the diffusion equation can be reduced to the second-order ordinary differential equation. Since $ζ = x / \sqrt{4 D t}$ , $\partial ζ / \partial t = - ζ / (2 t)$ . By the chain rule, $C_{t} (x, t)$ is given by

C_{t} (x, t) = \frac{\partial C (x, t)}{\partial t} = \frac{d f (ζ)}{d ζ} \frac{\partial ζ}{\partial t} = - \frac{ζ}{2 t} f^{'} (ζ)

(7)

Similarly, since $\partial ζ / \partial x = 1 / \sqrt{4 D t}$ , $C_{x x} (x, t)$ is given by

C_{x x} (x, t) = \frac{\partial}{\partial x} [\frac{\partial C (x, t)}{\partial x}] = \frac{\partial}{\partial x} [\frac{d f (ζ)}{d ζ} \frac{\partial ζ}{\partial x}] = \frac{1}{\sqrt{4 D t}} \frac{d f^{'} (ζ)}{d ζ} \frac{\partial ζ}{\partial x} = \frac{1}{4 D t} f^{″} (ζ)

(8)

Combining Equations (7) and (8) with Equation (1) gives

C_{t} (x, t) - D C_{x x} (x, t) = - \frac{ζ}{2 t} f^{'} (ζ) - \frac{D}{4 D t} f^{″} (ζ) = 0 (9)

2 ζ f^{'} (ζ) + f^{″} (ζ) = 0

(10)

The second ordinary differential $2 ζ f^{'} (ζ) + f^{″} (ζ) = 0$ can be solved as follows

\frac{f^{″} (ζ)}{f^{'} (ζ)} = \frac{d}{d ζ} \ln [f^{'} (ζ)] = - 2 ζ

(11)

\int d \ln [f^{'} (ζ)] = \int - 2 ζ d ζ \to \ln [f^{'} (ζ)] = - ζ^{2} + c_{1} (12)

where

c_{1}

is a constant of integration. By exponentiating,

f^{'} (ζ)

is given by

f^{'} (ζ) = e^{- ζ^{2} + c_{1}} = e^{c_{1}} e^{- ζ^{2}} = c_{2} e^{- ζ^{2}}

(13)

where

c_{2} = e^{c_{1}}

is constant. By integrating Equation (13),

f (ζ)

is given by

f (ζ) = \int_{0}^{ζ} c_{2} e^{- ξ^{2}} d ξ + c_{3}

(14)

where

c_{3}

is another constant of integration. Thus, the special solution of the form

C (x, t) = f (x / \sqrt{4 D t})

can be obtained as follows

C (x, t) = f (\frac{x}{\sqrt{4 D t}}) = f (ζ) = c_{2} \int_{0}^{ζ} e^{- ξ^{2}} d ξ + c_{3} = c_{2} \int_{0}^{\frac{x}{\sqrt{4 D t}}} e^{- ξ^{2}} d ξ + c_{3}

(15)

Since the boundary condition is $C (0, t) = C_{1}$ for $x = 0 and t > 0$ , then $c_{3} = C_{1}$ is as follows

C (0, t) = c_{2} \int_{0}^{\frac{0}{\sqrt{4 D t}}} e^{- ξ^{2}} d ξ + c_{3} = c_{2} \int_{0}^{0} e^{- ξ^{2}} d ξ + c_{3} = c_{3} = C_{1} (16)

(16)

Since the initial condition is $C (x, 0) = 0$ for $t = 0 and 0 < x < \infty$ , $\lim_{t \to 0} x / \sqrt{4 D t} = \infty$ by taking the limit $t \to 0$ , and $\int_{0}^{\infty} e^{- ξ^{2}} d ξ = \sqrt{π} / 2$ by the Gaussian integral. Then, $c_{2}$ is given by

\lim_{t \to 0} C (x, t) = c_{2} \int_{0}^{\infty} e^{- ξ^{2}} d ξ + C_{1} = c_{2} \frac{\sqrt{π}}{2} + C_{1} = 0 \to c_{2} = - C_{1} \frac{2}{\sqrt{π}}

(17)

Therefore, $C (x, t)$ of Equation (15) is given by

C (x, t) = - C_{1} \frac{2}{\sqrt{π}} \int_{0}^{\frac{x}{\sqrt{4 D t}}} e^{- ξ^{2}} d ξ + C_{1}

(18)

The rate of transfer, $F (x, t)$ , is obtained by differentiating $C (x, t)$ of Equation (18) with respect to $x$ as follows

F (x, t) = - D \frac{\partial C (x, t)}{\partial x} = - D \frac{\partial}{\partial x} (- C_{1} \frac{2}{\sqrt{π}} \int_{0}^{\frac{x}{\sqrt{4 D t}}} e^{- ξ^{2}} d ξ + C_{1})

(19)

Let $ξ = x / \sqrt{4 D t}$ , then $d ξ = d x / \sqrt{4 D t}$ . Thus, $F (x, t)$ is given by

F (x, t) = D \frac{\partial}{\partial x} (C_{1} \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- {(\frac{χ}{\sqrt{4 D t}})}^{2}} \frac{1}{\sqrt{4 D t}} d χ) = C_{1} \frac{\sqrt{D}}{\sqrt{π t}} e^{- \frac{x^{2}}{4 D t}}

(20)

The rate of transfer at $x = 0$ for the semi-infinite medium becomes

F (0, t) = C_{1} \frac{\sqrt{D}}{\sqrt{π t}}

(21)

when

C (x, 0) = 0

and

C (0, t) = C_{1}

, the total amount

Q

taken up by the medium in time

t

can be obtained by integrating

F (0, t)

with respect to

t

from

0

t

as follows¹⁵

Q = \int_{0}^{t} C_{1} \frac{\sqrt{D}}{\sqrt{π τ}} d τ = C_{1} \frac{\sqrt{D}}{\sqrt{π}} {[2 \sqrt{τ}]}_{0}^{t} = C_{1} \frac{2 \sqrt{D}}{\sqrt{π}} \sqrt{t}

(22)

The slope of the linear function of $Q$ with respect to $\sqrt{t}$ is equal to the differential coefficient of $Q$ with respect to $\sqrt{t}$ as follows

\frac{dQ}{d \sqrt{t}} = C_{1} \frac{2 \sqrt{D}}{\sqrt{π}}

(23)

Since $Q$ corresponds to the total amount of dye that has passed through the first-layer surface, let us put $Q$ as $Q_{1 \to n}$ and $D$ as $D_{1 \to n}$ . Then $D_{1 \to n}$ is given by

D_{1 \to n} = \frac{π}{4 {(C_{1})}^{2}} {(\frac{d Q_{1 \to n}}{d \sqrt{t}})}^{2}

(24)

Determination of constant first-layer surface concentration

In order to obtain the diffusion coefficient $D_{1 \to n}$ , the first-layer surface concentration $C_{1}$ must be determined first. Table 1 presents the mean dye concentration ( ${\bar{C}}_{i}$ ) of each layer for the diffusion of disperse dye in paste into PET film by treating it at 190°C for various heat treatment times using the film-roll method.

Table 1.

Mean dye concentration of each layer over various heat treatment times at 190°C

${\bar{C}}_{i} (mg \cdot {cm}^{- 3})$	15 min $(3.8730)$	30 min $(5.4772)$	60 min $(7.7460)$	120 min $(10.954)$	180 min $(13.416)$	240 min $(15.492)$
${\bar{C}}_{1}$	28.0	38.3	59.3	76.9	77.7	78.0
${\bar{C}}_{2}$	9.6	16.6	26.0	43.1	53.1	61.2
${\bar{C}}_{3}$	–	5.8	12.6	22.7	33.5	44.8
${\bar{C}}_{4}$	–	–	5.0	12.8	21.5	30.7
${\bar{C}}_{5}$	–	–	–	7.8	13.1	21.0
${\bar{C}}_{6}$	–	–	–	3.9	9.2	12.5
${\bar{C}}_{7}$	–	–	–	–	5.4	7.5
${\bar{C}}_{8}$	–	–	–	–	3.4	3.9
$M = \sum_{i = 1}^{8} \bar{C_{i}} (mg \cdot {cm}^{- 3})$	37.6	60.7	102.9	167.2	216.9	259.6
$Q_{1 \to n} = ε M (mg \cdot {cm}^{- 2})$	0.1504	0.2428	0.4116	0.6688	0.8676	1.0384
$Q_{2 \to n} = ε \sum_{i = 2}^{8} \bar{C_{i}} (mg \cdot {cm}^{- 2})$	0.0384	0.0896	0.1744	0.3612	0.5568	0.7264

The value of $Q_{1 \to n} = ε \sum_{i = 1}^{n} {\bar{C}}_{i}$ obtained by multiplying $M = \sum_{i = 1}^{n} {\bar{C}}_{i}$ by the thickness $ε (0.004 cm)$ of the layer is the total amount of dye that has passed through the first-layer surface whose concentration $C_{1}$ is constant. The value of $Q_{2 \to n} = ε \sum_{i = 2}^{n} {\bar{C}}_{i}$ is the total amount of dye that has passed through the second-layer surface whose concentration $C (40, t) = C_{x = 40}$ varies over time, where $n$ is the number of the last layer diffused.¹²

Figure 2 shows the distance–concentration distributions according to the heat treatment times for the diffusion at 190°C under the same condition as the infinite dye-bath and the steady-state concentration distribution deduced from the diffusion for 240 minutes. As time increased, ${\bar{C}}_{1}$ increased significantly until 120 minutes. However, the values of ${\bar{C}}_{1}$ for 120 and 180 minutes are almost the same as ${\bar{C}}_{1}$ for 240 minutes. Since ${\bar{C}}_{1}$ for 240 minutes was estimated to satisfy the steady-state condition, the steady-state straight line is also estimated to pass through ${\bar{C}}_{1}$ for 240 minutes.

Figure 2.

Concentration–distance curves over time and steady-state concentration distribution for diffusion at 190°C.

Assuming that the concentration distribution for the diffusion at 190°C for 240 minutes satisfies the steady-state condition approximately and the steady-state straight line passes through ${\bar{C}}_{1}$ for 240 minutes, the total amount of dye that has passed through the surface must be equal to the area of a triangle with a base of distance $l$ and a height of $C_{1}$ , that is, $ε M = C_{1} l / 2$ , where $ε = 40$ µm.¹⁰ Then $C_{1}$ and the linear function $C (x)$ of the steady-state straight line are given by

C_{1} = \frac{2}{l} ε M = \frac{80}{l} \sum_{i = 1}^{n} \bar{C_{i}} and C (x) = - \frac{C_{1}}{l} x + C_{1} (25)

Since ${\bar{C}}_{1}$ is the concentration at $x = 20$ µm, ${\bar{C}}_{1}$ is given by

{\bar{C}}_{1} = C (20) = - \frac{C_{1}}{l} \times 20 + C_{1} = - \frac{1600 M}{l^{2}} + \frac{80 M}{l}

(26)

Converting Equation (26) into a quadratic equation for $l$ yields the value of $l$ as follows

\begin{array}{l} l^{2} - \frac{80 M}{{\bar{C}}_{1}} l + \frac{1600 M}{{\bar{C}}_{1}} = 0 and \\ l = \frac{\frac{80 M}{{\bar{C}}_{1}} + \sqrt{{(\frac{80 M}{{\bar{C}}_{1}})}^{2} - 4 \frac{1600 M}{{\bar{C}}_{1}}}}{2} \end{array}

(27)

Therefore, $C_{1}$ can be obtained by $l$ and $C_{1} = 2 ε M / l$ . In the steady-state concentration distribution for the diffusion at 190°C, $l$ is 244.5 µm and $C_{1}$ is $84.9 mg \cdot {cm}^{- 3}$ . Considering that the first-layer mean concentrations ${\bar{C}}_{1}$ for heat treatment times less than 120 minutes are very small, it is estimated that $84.9 mg \cdot {cm}^{- 3}$ obtained from the steady-state distribution is more suitable for $C_{1}$ than $88.6 mg \cdot {cm}^{- 3}$ obtained by the $C$ -intercept of the quadratic regression equation for 240 minutes.

Table 2 presents the mean dye concentration ( ${\bar{C}}_{i}$ ) of each layer and the total amounts of $Q_{1 \to n}$ and $Q_{2 \to n}$ for the diffusion of disperse dye into PET film by treating it at 180°C for various times. Figure 3 shows the distance–concentration distributions at 180°C and the steady-state concentration distribution deduced from the diffusion for 240 minutes. As at 190°C, ${\bar{C}}_{1}$ for 180 minutes is almost the same as ${\bar{C}}_{1}$ for 240 minutes, so ${\bar{C}}_{1}$ for 240 minutes is expected to satisfy the steady-state condition. Therefore, the steady-state straight line with $l$ of $170.1 μ m$ and $C_{1}$ of $57.1 mg \cdot {cm}^{- 3}$ is expected to pass through ${\bar{C}}_{1}$ for 240 minutes.

Table 2.

Mean dye concentration of each layer over various heat treatment times at 180°C

${\bar{C}}_{i} (mg \cdot {cm}^{- 3})$	30 min $(5.4772)$	60 min $(7.7460)$	120 min $(10.954)$	180 min $(13.416)$	240 min $(15.492)$
${\bar{C}}_{1}$	25.9	34.9	45.4	49.5	50.4
${\bar{C}}_{2}$	6.7	13.9	21.0	28.5	34.0
${\bar{C}}_{3}$	–	3.9	10.5	13.8	20.2
${\bar{C}}_{4}$	–	–	5.4	7.5	11.6
${\bar{C}}_{5}$	–	–	–	3.5	5.2
$M = \sum_{i = 1}^{5} {\bar{C}}_{i} (mg \cdot {cm}^{- 3})$	32.6	52.7	82.3	102.8	121.4
$Q_{1 \to n} = ε M (mg \cdot {cm}^{- 2})$	0.1304	0.2108	0.3292	0.4112	0.4856
$Q_{2 \to n} = ε \sum_{i = 2}^{5} {\bar{C}}_{i} (mg \cdot {cm}^{- 2})$	0.0268	0.0712	0.1476	0.2132	0.2840

Figure 3.

Concentration–distance curves over time and steady-state concentration distribution for diffusion at 180°C.

Table 3 shows the mean dye concentration ( ${\bar{C}}_{i}$ ) of each layer and the total amounts of $Q_{1 \to n}$ and $Q_{2 \to n}$ for the diffusion of disperse dye into PET film by treating it at 170°C for various times. Figure 4 shows the distance–concentration distributions at 170°C and the steady-state concentration distribution deduced from the diffusion for 300 minutes. As at 190°C and 180°C, ${\bar{C}}_{1}$ increased significantly until 180 minutes, but ${\bar{C}}_{1}$ for 240 minutes is almost the same as ${\bar{C}}_{1}$ for 300 minutes. Therefore, the steady-state straight line with $l$ of $147.7 μ m$ and $C_{1}$ of $37.6 mg \cdot {cm}^{- 3}$ is also expected to pass through ${\bar{C}}_{1}$ for 300 minutes.

Table 3.

Mean dye concentration of each layer over various heat treatment times at 170°C

${\bar{C}}_{i} (mg \cdot {cm}^{- 3})$	60 min $(7.7460)$	120 min $(10.954)$	180 min $(13.416)$	240 min $(15.492)$	300 min $(17.321)$
${\bar{C}}_{1}$	21.5	26.9	30.8	32.2	32.5
${\bar{C}}_{2}$	7.8	10.2	11.6	16.6	20.0
${\bar{C}}_{3}$	3.3	4.7	8.3	11.1
${\bar{C}}_{4}$			1.8	3.6	5.8
$M = \sum_{i = 1}^{4} {\bar{C}}_{i} (mg \cdot {cm}^{- 3})$	29.3	40.4	48.9	60.7	69.4
$Q_{1 \to n} = ε M (mg \cdot {cm}^{- 2})$	0.1172	0.1616	0.1956	0.2428	0.2776
$Q_{2 \to n} = ε \sum_{i = 2}^{4} {\bar{C}}_{i} (mg \cdot {cm}^{- 2})$	0.0312	0.0540	0.0724	0.1140	0.1476

Figure 4.

Concentration–distance curves over time and steady-state concentration distribution for diffusion at 180°C.

Diffusion coefficient by constant first-layer surface concentration

According to Equation (22), $Q_{1 \to n}$ is a linear function of $\sqrt{t}$ and the straight line of $Q_{1 \to n}$ with respect to $\sqrt{t}$ must pass through $(0, 0)$ of $(\sqrt{t}, Q_{1 \to n})$ . Figure 5 shows the linear regression lines for plots of $Q_{1 \to n}$ against $\sqrt{t}$ according to the data in Tables 1 –3. Of the two plots at each temperature, one is the linear regression line for data containing $(0, 0)$ and the other is the linear regression line for data not containing $(0, 0)$ . The slope of the linear regression line of $Q_{1 \to n}$ with respect to $\sqrt{t}$ is equal to the differential coefficient $d Q_{1 \to n} / d \sqrt{t}$ of Equation (23). Thus, the diffusion coefficients $D_{1 \to n}$ can be obtained by $C_{1}$ and $d Q_{1 \to n} / d \sqrt{t}$ of Equation (24). Table 4 presents the correlation coefficients $r$ , the slopes $dQ / d \sqrt{t}$ , and the diffusion coefficients $D_{1 \to n}$ for plots of total amounts against square roots of times in Figure 5.

Figure 5.

Plots of total amounts against square roots of treatment times for diffusions at 170°C, 180°C, and 190°C.

Table 4.

Correlation coefficients $(r)$ , slopes $(dQ / d \sqrt{t})$ , and $D_{1 \to n}$ for plots of total amounts against square roots of times

Linear regression	190°C		180°C		170°C
Linear regression	Containing $(0, 0)$ of $(\sqrt{t}, Q_{1 \to n})$	Not containing $(0, 0)$	Containing $(0, 0)$ of $(\sqrt{t}, Q_{1 \to n})$	Not containing $(0, 0)$	Containing $(0, 0)$ of $(\sqrt{t}, Q_{1 \to n})$	Not containing $(0, 0)$
$r$	$0.9913$	$0.9992$	$0.9958$	$0.9999$	$0.9948$	$0.9959$
$dQ / d \sqrt{t}$	$0.06971$	$0.07740$	$0.03198$	$0.03546$	$0.01616$	$0.01749$
$D_{1 \to n}$ $({cm}^{2} \min^{- 1})$	$5.2923 \times 10^{- 7}$	$6.5243 \times 10^{- 7}$	$2.4624 \times 10^{- 7}$	$3.0274 \times 10^{- 7}$	$1.4500 \times 10^{- 7}$	$1.6985 \times 10^{- 7}$

In the diffusion at 190°C, the correlation coefficient of linear regression for data not containing $(0, 0)$ is 0.9992, so the linearity of $Q_{1 \to n}$ with respect to $\sqrt{t}$ for data not containing $(0, 0)$ is superior. On the other hand, the correlation coefficient of linear regression for data containing $(0, 0)$ is 0.9913, so the linearity of $Q_{1 \to n}$ with respect to $\sqrt{t}$ for data containing $(0, 0)$ is inferior to that for data not containing $(0, 0)$ . However, the linear regression line for data containing $(0, 0)$ is more consistent with Equation (22) of a linear function passing through $(0, 0)$ . The diffusion coefficient for data not containing $(0, 0)$ is $6.5243 \times 10^{- 7} {cm}^{2} \min^{- 1}$ and that for data containing $(0, 0)$ is $5.2923 \times 10^{- 7} {cm}^{2} \min^{- 1}$ .

In the diffusion at 180°C, the correlation coefficient of linear regression for data not containing $(0, 0)$ is 0.9999, so the linearity of $Q_{1 \to n}$ with respect to $\sqrt{t}$ for data not containing $(0, 0)$ is superior. However, the linear regression line for data containing $(0, 0)$ with the correlation coefficient of 0.9958 is also good in linearity. The diffusion coefficient $D_{1 \to n}$ for data not containing $(0, 0)$ is $3.0274 \times 10^{- 7} {cm}^{2} \min^{- 1}$ and $D_{1 \to n}$ for data containing $(0, 0)$ is $2.4624 \times 10^{- 7} {cm}^{2} \min^{- 1}$ .

In the diffusion at 170°C, both the linear regression line for data not containing $(0, 0)$ with the correlation coefficient of 0.9959 and that for data containing $(0, 0)$ with the correlation coefficient of 0.9948 have good linearity. The diffusion coefficient for data not containing $(0, 0)$ is $1.6985 \times 10^{- 7} {cm}^{2} \min^{- 1}$ and that for data containing $(0, 0)$ is $1.4500 \times 10^{- 7} {cm}^{2} \min^{- 1}$ .

Diffusion coefficient by time-varying second-layer surface concentration for comparison

In a previous study on the diffusion of the time-varying surface concentrations, the solution of the diffusion equation for $C (0, t) = k \sqrt{t}$ ( $k$ is the proportional constant) deduced by applying the Laplace transform is given by¹²

C (x, t) = k \sqrt{π t} \int_{\frac{x}{\sqrt{4 D t}}}^{\infty} erfc (ζ) d ζ

(28)

Figure 6 shows the surface with constant $C_{1}$ and the surface with time-varying $C_{x = 40}$ in the film layers and concentration distribution using the diffusion at 180°C. If the second-layer surface is the surface from which diffusion begins and the second-layer surface concentration $C_{x = 40}$ varies in proportion to the square root of time, the delay time $t_{0}$ is the time taken for a dye in the paste to pass the first-layer surface and reach the second-layer surface. Then, the diffusion time is $t - t_{0}$ , the diffusion distance is $x - 40$ , and the boundary condition is $C_{x = 40} = k \sqrt{t - t_{0}} (t > t_{0})$ . Thus, $C (x - 40, t - t_{0})$ is given by

C (x - 40, t - t_{0}) = k \sqrt{π (t - t_{0})} \int_{\frac{x - 40}{\sqrt{4 D (t - t_{0})}}}^{\infty} erfc (ζ) d ζ (29)

Figure 6.

Surface with $C_{1}$ and surface with $C_{x = 40}$ in film layers and concentration distribution for diffusion at 180°C.

The rate of transfer $F (x, t)$ also shifts to $F (x - 40, t - t_{0})$ , which is obtained by differentiating $C (x - 40, t - t_{0})$ with respect to $x$ as follows

F (x - 40, t - t_{0}) = - D \frac{\partial C (x - 40, t - t_{0})}{\partial x} = \frac{k \sqrt{π D}}{2} erfc (\frac{x - 40}{\sqrt{4 D (t - t_{0})}}) (30)

Since $erfc (0) = 1$ , the rate of transfer at $x = 40$ is given by

F (0, t - t_{0}) = \frac{k \sqrt{π D}}{2}

(31)

The total amount $Q_{2 \to n}$ is obtained by integrating Equation (31) with respect to $t$ from $0$ to $t - t_{0}$ as follows¹⁶

Q_{2 \to n} = ε \sum_{i = 2}^{n} {\bar{C}}_{i} = \int_{0}^{t - t_{0}} \frac{k \sqrt{π D}}{2} d τ = \frac{k \sqrt{π D}}{2} (t - t_{0})

(32)

The delay time $t_{0}$ corresponds to the $t$ -axis intercept in the regression line for the plot of $Q_{2 \to n}$ against $t$ .

Let us put $D$ as $D_{2 \to n}$ . Since $d Q_{2 \to n} / d t$ is $k \sqrt{π D} / 2$ , $D_{2 \to n}$ is given by

D_{2 \to n} = \frac{4}{π k^{2}} {(\frac{d Q_{2 \to n}}{d t})}^{2}

(33)

Figure 7 shows the plots of $Q_{2 \to n}$ against $t$ for the diffusions at 190°C, 180°C, and 170°C according to the data in Tables 1 –3 with their regression lines. The delay time $t_{0}$ was obtained by the slope and the $Q_{2 \to n}$ -intercept in the equation of linear regression line corresponding to Equation (32). The values of $C_{x = 40}$ according to $\sqrt{t - t_{0}}$ are presented in Table 5 to find the proportional constant $k$ .

Figure 7.

Plots of total amounts for time-varying second-layer surface concentrations against heat treatment times.

Table 5.

Time-varying second-layer surface concentrations according to square root of treatment time

190°C $(t_{0} = 1.2 \min)$			180°C $(t_{0} = 2.9 \min)$			170°C $(t_{0} = 10.6 \min)$
$t - t_{0}$	$\sqrt{t - t_{0}}$	$C_{x = 40} (mg \cdot {cm}^{- 3})$	$t - t_{0}$	$\sqrt{t - t_{0}}$	$C_{x = 40} (mg \cdot {cm}^{- 3})$	$t - t_{0}$	$\sqrt{t - t_{0}}$	$C_{x = 40} (mg \cdot {cm}^{- 3})$
13.8	3.7148	18.8	–	–	–	–	–	–
28.8	5.3666	26.7	27.1	5.2058	16.3	–	–	–
58.8	7.6681	41.8	57.1	7.5565	23.6	49.4	7.0285	14.7
118.8	10.900	59.4	117.1	10.821	32.4	109.4	10.459	18.0
178.8	13.372	64.4	177.1	13.308	38.3	169.4	13.015	20.6
238.8	15.453	69.1	237.1	15.398	41.6	229.4	15.146	23.9
–	–	–	–	–	–	289.4	17.012	25.8

Figure 8 shows the linear regression lines for the plots of $C_{x = 40}$ against $\sqrt{t - t_{0}}$ . The proportional constants $k$ of $C_{x = 40} = k \sqrt{t - t_{0}}$ were determined by the slope of the linear regression lines. The diffusion coefficients $D_{2 \to n}$ were determined by $d Q_{2 \to n} / d t$ in Figure 7 and $k$ according to Equation (33).

Figure 8.

Plots of time-varying second-layer surface concentrations against square roots of diffusion times.

Arrhenius plot and activation energy

The dependence of the diffusion coefficient on temperature is given by the Arrhenius equation as follows

D = A e^{- \frac{E}{RT}}

(34)

where

A

is a temperature-independent constant for the diffusion,

E

is the activation energy,

T

is the absolute temperature, and

R

is the gas constant (1.987

cal \cdot K^{- 1} \cdot {mol}^{- 1}

).¹⁷ The Arrhenius equation gives the quantitative basis of the relationship between the diffusion coefficient and the activation energy. The Arrhenius plot is often used to analyze the effect of temperature on the rate of diffusion. For a single rate-limited thermally activated process, the Arrhenius plot gives a straight line that is the linear function of

- \log D

with respect to

T^{- 1}

by taking the common logarithm of Equation (34) as follows¹⁸

- \log D = \frac{E \log e}{R} (T^{- 1}) - \log A

(35)

The activation energy $E$ can be obtained by the slope of the linear regression line for the Arrhenius plot because the slope of Equation (35) is equal to $d (- \log D) / d (T^{- 1})$ as follows

\frac{d (- \log D)}{d (T^{- 1})} = \frac{E \log e}{R}

(36)

The closer the correlation coefficient of linear regression for the Arrhenius plot is to 1, the clearer it becomes that the diffusion coefficient for diffusion at a specified temperature is constant and the reliability of the diffusion coefficients increases. In this study, the linearity of linear regression for the Arrhenius plot was used to prove the reliability of the diffusion coefficients obtained by Equation (22) and to confirm that the diffusion coefficient for diffusion at a specified temperature is constant. The activation energies were also used to prove the reliability of the diffusion coefficients by comparing them with that for the diffusion coefficients by the time-varying second-layer surface concentrations.

The diffusion coefficients $D_{1 \to n}$ and $D_{2 \to n}$ for the diffusions at 170°C, 180°C, and 190°C with their common logarithms are presented in Table 6 to obtain the Arrhenius plots for $D_{1 \to n}$ and $D_{2 \to n}$ .

Table 6.

Diffusion coefficients for constant first-layer surface concentration and time-varying second-layer surface concentration

$°C$	$T$ $(K)$	$T^{- 1} \times 10^{3}$	Data not containing $(0, 0)$		Data containing $(0, 0)$ of $(\sqrt{t}, Q_{1 \to n})$		$D_{2 \to n} ({cm}^{2} \min^{- 1})$	$- \log D_{2 \to n}$
$°C$	$T$ $(K)$	$T^{- 1} \times 10^{3}$	$D_{1 \to n} ({cm}^{2} \min^{- 1})$	$- \log D_{1 \to n}$	$D_{1 \to n} ({cm}^{2} \min^{- 1})$	$- \log D_{1 \to n}$	$D_{2 \to n} ({cm}^{2} \min^{- 1})$	$- \log D_{2 \to n}$
190	463	2.160	$6.5243 \times 10^{- 7}$	6.1855	$5.2923 \times 10^{- 7}$	$6.2764$	$5.4868 \times 10^{- 7}$	6.2607
180	453	2.208	$3.0274 \times 10^{- 7}$	6.5189	$2.4624 \times 10^{- 7}$	$6.6086$	$2.4459 \times 10^{- 7}$	6.6116
170	443	2.257	$1.6985 \times 10^{- 7}$	6.7699	$1.4500 \times 10^{- 7}$	$6.8386$	$1.4553 \times 10^{- 7}$	6.8370

Figure 9 shows the Arrhenius plots for $D_{1 \to n}$ of data containing $(0, 0)$ of $(\sqrt{t}, Q_{1 \to n})$ , for $D_{1 \to n}$ of data not containing $(0, 0)$ , and for $D_{2 \to n}$ with their linear regression lines. The correlation coefficient of linear regression for data not containing $(0, 0)$ is 0.9962 and that for data containing $(0, 0)$ is 0.9939, so both linear regression lines have good linearity. Their good linearity proved the reliability of the diffusion coefficients $D_{1 \to n}$ obtained by Equation (22) and confirmed that the diffusion coefficient for diffusion at a specified temperature is constant. The values of $D_{2 \to n}$ are similar to those of $D_{1 \to n}$ for data containing $(0, 0)$ , as shown in Table 6, so the linear regression line for the Arrhenius plot of $D_{2 \to n}$ with the correlation coefficient of 0.9914 is very similar to that for the Arrhenius plot of $D_{1 \to n}$ for data containing $(0, 0)$ . The activation energy $E_{2 \to n}$ obtained by the slope of the linear regression line of $- \log D_{2 \to n}$ with respect to $T^{- 1}$ is $27.2 kcal \cdot {mol}^{- 1}$ , which is similar to $E_{1 \to n}$ of $27.6 kcal \cdot {mol}^{- 1}$ for data not containing $(0, 0)$ and $E_{1 \to n}$ of $26.5 kcal \cdot {mol}^{- 1}$ for data containing $(0, 0)$ . The reliability of $D_{1 \to n}$ was also proved by the good similarity between $E_{1 \to n}$ for $D_{1 \to n}$ and $E_{2 \to n}$ for $D_{2 \to n}$ . In particular, $D_{1 \to n}$ for data containing $(0, 0)$ is judged to be more reliable than $D_{1 \to n}$ for data not containing $(0, 0)$ due to the good consistency with $D_{2 \to n}$ for the time-varying second-layer surface concentration.

Figure 9.

Arrhenius plots of diffusion coefficients for constant surface concentration and time-varying surface concentration.

Conclusions

In order to find an equation for obtaining a reliable diffusion coefficient for the sublimation diffusion of disperse dye in paste into PET film using the film-roll method, an equation where the total amount of dye passed through the first-layer surface is a linear function of the square root of time was derived from the solution of the diffusion equation with the constant surface concentration. The constant surface concentration was obtained by the steady-state straight line at a sufficiently long heat treatment time. The diffusion coefficients obtained by the constant first-layer surface concentration and the slope of the linear regression line for the plot of the total amount against the square root of time have been proven highly reliable by the good linearity of linear regression for the Arrhenius plot. The activation energy for the diffusion coefficients obtained by the equation with the constant first-layer surface concentration was similar to that obtained by another equation with the time-varying second-layer surface concentration. The diffusion coefficient $D_{1 \to n}$ for data containing $(0, 0)$ of $(\sqrt{t}, Q_{1 \to n})$ was judged to be more reliable than $D_{1 \to n}$ for data not containing $(0, 0)$ due to the good consistency with $D_{2 \to n}$ .

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Geon Yong Park

References

Datyner

Delaney

Jima

The interaction between acid dyes and nonionic surfactants and its effect on sorption and diffusion behavior: Part III: diffusion of dyes into polyamide 6 films. Text Res J 1973; 43: 48–53.

Geonyong

Jinwoo

Hydrophilic graft copolymerization of nylon 6 with acrylamide(II). J Kor Fiber Soc 1988; 25: 66–71.

Sicardi

Manna

Banchero

Diffusion of disperse dyes in PET films during impregnation with a supercritical fluid. J Supercrit Fluid 2000; 17: 187–194.

Sekido

Matsui

Study on vat dyeing. Sen-i Gakkaishi 1964; 20: 778–783.

Jones

Leung

TSM.

Some fundamental aspects of transfer printing. J Soc Dyers Colour 1974; 90: 286–290.

Geonyong

Factors affecting the absorption and diffusion of disperse dye in print paste for polyester film. J Kor Soc Dyers Finishers 2007; 19: 21–27.

Geonyong

Diffusion behaviors of disperse dye to PET film from print paste using film-roll method. Text Sci Eng 2017; 54: 125–130.

Geonyong

Diffusion coefficient for sublimation diffusion of disperse dye using error function. Int J Innovative Technol Exploring Eng 2019; 8: 37–42.

Geonyong

Diffusion coefficient calculated by complementary error function for the sublimation diffusion of disperse dye. J Eng Fibers Fabrics 2019; 14: 1–9.

10.

Geonyong

Study on calculation of diffusion coefficient for sublimation diffusion of disperse dye using Fourier series. Fiber Polym 2020; 21: 538–547.

11.

Geonyong

Diffusion coefficient calculated by time-lag using film-roll method. Text Res J 2022; 92: 91–102.

12.

Geonyong

Diffusion coefficient for diffusion of time-varying surface concentration by the film-roll method. Text Res J 2022; 92: 1891–1908.

13.

Madan

Khan

AH.

Determination of dye on textile fibers. Part I: disperse dye on polyethylene terephthalate. Text Res J 1978; 48: 481–486.

14.

Cain JW and Reynolds AM. Ordinary and partial differential equations. Richmond: Virginia Commonwealth University Press, 2010, pp.266–268.

15.

Crank

The mathematics of diffusion. Oxford: Clarendon Press, 1975, p.32.

16.

Crank

The mathematics of diffusion. Oxford: Clarendon Press, 1975, p.33.

17.

Laidler

KJ.

The development of the Arrhenius equation. J Chem Educ 1988; 61: 494–498.

18.

Truhlar

Kohen

Convex Arrhenius plots and their interpretation. Proc Natl Acad Sci 2001; 98: 848–851.