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Polydispersity of macromolecular solutions and colloids in electro-optics
Colloids and Surfaces A: Physicochemical and Engineering Aspects 148 (1999) 29–34
Polydispersity of macromolecular solutions and colloids in electro-optics L.K. Babadzanjanz, M.L. Bregman, A.A. Trusov, V.V. Vojtylov * Faculty of Applied Mathematics and Processes of Management, Faculty of Physics, St Petersburg State University, Ulianovskaia 1, Petrodvoretz, St Petersburg 198904, Russia
Abstract This article analyzes the problems of polydispersity in electro-optics. It examines the application of the regularisation technique in order to determine distribution functions of colloidal particles and macromolecules on electric dipole moments, on polarisability anisotropy values, and on rotary friction coefficients. It also includes two additions to the regularisation technique. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Conductivity anisotropy; Dipole moment; Distribution function; Electro-optical methods; Polarisability
where
1. Introduction In what follows we consider methods of investigating the particles of liquid systems. These particles are macromolecules in solutions and colloid particles in disperse systems. The characteristics of these particles are the anisotropy of polarisability c, the permanent dipole moment m and the rotary diffusion coefficient D. These characteristics can be determined by well-known electro-optical methods [1], if the system in question is monodisperse. These include birefringence, dichroism, anisotropy of conductivity and permittivity [2,3]. Mainly, we will consider the polydisperse systems, but we will begin with monodisperse ones. The effects we discuss can be determined for monodisperse systems by the following equation: A=A W max * Corresponding author. Fax: +7 812 114 3387; e-mail: [email protected]
(1)
W=
P
p 3 cos2 h−1 0
2
F(h) sin h dh
(2)
The quantity W is a factor of particle orientation, h is the angle between the field direction and the symmetry axis of the particle or macromolecule and F(h) is the angular distribution function. The function F(h) and the factor W depend on the kind of electric field applied to the system, the strength E of the field, the frequency v, the time t and the particle characteristics c, m, D. The quantity A is the magnitude of electro-orientational effect. Electro-orientational effects consist of: birefringence, dichroism, conductivity anisotropy, and permittivity anisotropy. The quantity A is equal max to A if the orientation of the particles is saturated. To study polydisperse systems we will also determine the distribution function of c, m and D. The number distribution functions show the dependences between the values c, m and D. These
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L.K. Babadzanjanz et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 148 (1999) 29–34
dependences may be used when studying such complicated systems as solutions of DNA, polypeptides and suspensions of biological cells. To simplify the problem, the angular distribution function F(h) will depend only on the characteristic m, c or D. The distribution function of the maximum effect A on parameters of particles of polydisperse systems can be determined as follows: ∂A ∂A ∂A f (c)¬ max , f ( m)¬ max , f (D)¬ max 1 2 3 ∂c ∂m ∂D (3) The general equation for these functions is:
P
A(y)=
b
K(y, j) f (j) dj
where c˜ ¬cE2 /2kT, k is the Boltzmann coefficient, 0 and T the absolute temperature. 1.2. Second example: the case of m Here we consider the distribution of permanent dipole moments. There are two subcases represented by different formulae. (a) E=E , y=E, j=m 0 1 3 coth( m˜ ) K(y, j)=W ( m˜ )¬1+ − (7) 2a m˜ 2 m˜ m˜ ¬
mE kT
(4) (b) V&D
a
Here: y=E,
E2 2
K(y, j)=W ( m˜ , c˜ )¬2 2b
1
, v, t; j=c, m, D,
D
f (j)=f (c), f ( m), f (D) (5) 1 2 3 The main problem of this work is that of solving the integral Eq. (4). The analytic formulae for the kernel K(y, j) depend on the different kinds of electrical fields applied to the system. Here we have to note that in all cases we consider the kernel as a function of one variable, namely y2j, yj, and y/j. Now let us consider some examples of such a kernel [4,5]. 1.1. First example: the case of c Let us consider the kernel of the integral equation for anisotropy polarisability distribution with the applied field being sine-shaped and of high frequency [2,6 ]: E=E sin(Vt), V&D 0 y=E2 /2, j=c 0 Here we have: K(y, j)=W (c˜ ) 1 3 1 ¬ x2 ec˜ x2 dx 2 0
P
P
c 0
×[u(x)+u(1−x2)] u(x)¬x3 ec˜ x2
C
P
dx 앀1−x2
1/앀2 1−2y2 0
앀1−y2
A
× coth(c˜ m˜ y)+coth c˜ m˜ 앀1−y2
BD
dy
(8)
(a) is the subcase where the permanent dipole moment is much bigger than the induced one. This happens if the system consists of polar macromolecules or particles [7]. (b) is the subcase when the field direction changes with high frequency V within the limits of a right angle [8]. In many situations the dependence of the kernel on c˜ is negligible because values of the function W of 2b high polarisability are close to values of the function W of low polarisability. 2b 1.3. Third example: the case of D
NP
1 0
ec˜ x2 dx−
1 2
(6)
The third example arises if we consider the rotation friction of the particles in two different subcases corresponding to different kinds of fields applied. (a) High field ( m˜ +c˜ &1). This is a subcase of
31
L.K. Babadzanjanz et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 148 (1999) 29–34
relaxation from the saturated orientation [1,5]:
2.1. Regularisation
y=t, j=D
This method works only for smooth functions. Let us begin with the main equation of the Tikhonov regularisation technique [10] to solving the ill-posed problem (4):
K(y, j)=W (t˜)¬e−t˜ 3 ˜t¬6Dt
(9)
(b) Low field (c˜ %1). This is the subcase of sinefields of high frequency with sine-shaped amplitude modulation of low frequency: E=E sin(Vt) sin(vt) 0 V&D, v∏V (10)
y=v, j=D
The value A of electro-orientational effect can be determined by the following equation [9]: A(v)=
c˜ 30
− v ˜¬
C
1+ 1
1+v ˜2
v ˜ 1+v ˜2
sin(2vt)
D
cos(2vt)
(11)
P
b a
˜ (j , j) f (j) dj −a K 1 1
P
˜ (j, j )¬ K 1
P
d
3D
c yµ[c, d ]
(12)
1 K(y, j)=W (v ˜ )¬ 5 1+v ˜2
(13)
2. Methods and results We will discuss techniques attaining to the following: $ regularisation; $ modification of the kernel; $ equations for derivatives of distribution functions.
D
−f (j) =b(j)
where a is a real constant, which can be calculated by various techniques [5,10]. The regularisation method requires that the solution of Eq. (14) belong to a class of smooth functions. This is natural in case of the physical problems under consideration. The algorithm of the regularisation technique could be briefly described as follows. Step 1. Through the numerical integration technique one obtains the functions:
b(j)¬
v ˜ K(y, j)=W (v ˜ )¬ 4 1+v ˜2
dj2
(14)
v
Sine-shaped and cosine-shaped terms can be measured separately in experiments and give the possibility of determining the following kernels:
C
d2f (j)
d c
K(y, j)K(y, j ) dy 1
(15)
A(y)K(y, j) dy (16)
Step 2. Eq. (14) can be reduced to a system of linear algebraic equations. By solving this system, one finds a numerical solution to the integral Eq. (4). The advantage of this technique is that it is not sensitive to the noise–signal ratio. The chief disadvantage of this technique is that it fails to resolve problems under consideration if multimodal solutions occur. Let us have a look at the result of the numerical simulation of the unimodal solution for kernel numbered (9). This is the case of relaxation, from the saturate orientation of macromolecules or particles. The results are shown in Fig. 1. As one can see, in case of bimodal function the result is much worse. The results are shown in Fig. 2. The other two techniques we have to discuss have been designed as an addition to regularisation techniques in order to improve the results for multimodal solutions.
32
L.K. Babadzanjanz et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 148 (1999) 29–34
we derive:
P
b A (y)= 1 K (y, j) f (j) dj d d a1 where
(17)
∂mA(y) A (y)¬ d ∂ym
(18)
∂mK(y, j) K (y, j)¬ d ∂ym
(19)
[a , b ]5[a, b] 1 1 Fig. 1. Unimodal distribution function. The solid curve is a model curve, the dashed curve is a solution with a 10% random error of A(y), the dot-dashed curve is a solution with a 20% random error of A(y), the value D corresponds to the maxi0 mum of f(D).
Consider two examples: (a) the kernel is given by Eq. (6); (b) the kernel is given by Eqs. (12) and (13). As one can see in Fig. 3(a–c), the graphics of the derivatives of the kernels are non-negligible in a more narrow range of the argument. If the derivative of A(y) is measured in an experiment then one can solve Eq. (17) using the regularisation technique for two-modal functions. It gives more accurate results because the new kernel is narrower than the original one. 2.3. Equations for derivative distribution functions Here we assume that the kernel is a function of one argument yj. Thus we consider the equation:
P
b
A(y)=
Fig. 2. Bimodal distribution function. The solid curve is a model curve, the dashed curve is a solution calculated by the regularisation technique, the value D corresponds to the first maximum 0 of f(D).
K(yj) f (j) dj
(20)
a
where K(y, j)=K(yj) Besides, we assume: f (m)(a)=f (m)(b)=0, m=0, 1, 2, …
(21)
2.2. Modification of the kernel The kernel modification method is based on the differentiation of the original Eq. (4). In such a way the kernel reduces to another one which changes in a more narrow gap [a , b ] than [a, b]; 1 1
Combining the procedures of differentiation and integration by parts, we derive the formulae: A (y)= m
P
b a
K(yj)jm f(m)(j) dj
(22)
L.K. Babadzanjanz et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 148 (1999) 29–34
33
where A (y)=A(y) 0
(23)
=(1−m)A −y∂A (y)/∂y, m=1, 2, … m+1 m m Eq. (22) has the same form as Eq. (4), and it is an integral equation on the product of jm and f (m)(j)¬∂m f/∂ym. If m=0 then Eq. (22) is the same as the original Eq. (20). For m greater than 0 it becomes the integral equation with respect to m multiplied by the derivative of order m with the kernel being the same. The scheme of the technique under consideration for m=1 is as follows. Step 1. We find the data: A
A(y ), A(y ), …, A(y ) (24) 1 2 k A (y ), A (y ), …, A (y ) (25) d 1 d 2 d k from the experiment, and then we get analytical approximations for A of y and its derivative. Step 2. Using the regularisation technique we get a numerical solution of the equations:
P
A (y)= 0
b
K(yj) f (j) dj
(26)
a
A (y)=A (y)−y 1 0
P
∂A (y) b 0 = K(yj)jf ∞(j) dj ∂y a (27)
namely the data: f (j ), f (j ), …, f (j ) (28) 1 2 n j f ∞(j ), j f ∞(j ), …, j f ∞(j ) (29) 1 1 2 2 n n Step 3. Using the results of Eqs. (28) and (29) we construct an analytical formula for f(j). For example, at Steps 1 and 3 we can use splines constructed by a least-squares approximation method. Fig. 3. Graphs of kernels and their derivatives. (a) The solid curve is function W (c˜ ), the dashed curve is function dW /dc˜ , 1 1 the dot-dashed curve is d2W /dc˜ 2. (b) The solid curve is func1 tion W (v ˜ ), the dashed curve is function dW /dv ˜ . (c) The solid 4 4 curve is function W (v ˜ ), the dashed curve is function dW /dv ˜. 5 5
3. Conclusion So, it is stated (Section 2.1) that the Tikhonov regularisation technique for solving the ill-posed
34
L.K. Babadzanjanz et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 148 (1999) 29–34
problems of polydispersity in electro-optics is effective if the solution (distribution function) occurs as a one-modal function and fails if it has two or more modes. To improve this method, one could use the two techniques suggested above (Sections 2.2 and 2.3). The first technique can be used to change the original ill-posed problem to a different ill-posed problem. Applying the regularisation technique to the latter problem, one gets a more accurate result because the domain of the new kernel is narrower than the original one. The second technique gives an opportunity to find the derivative of the solution of the original integral equation and, in particular, to recognize how many modes the solution has. Further modeling experiments are to be carried out in order to investigate these techniques more thoroughly.
Acknowledgment The authors thank the Russian Fond for Fundamental Investigations for financial support
(Grant RFFI 98-03-32713a and Grant RFFI 96-03-34310a).
References [1] H. Benoit, Ann. Phys. 6 (1951) 561. [2] S.Sp. Stoylov, V.N. Shilov, S.S. Dukhin, S. Sokerov, V. Petkanchin, Electrooptika kolloidov ( Electrooptics of Colloids), Naukova Dumka, Kiev, 1977 (in Russian). [3] H. Heller, Rev. Mod. Phys. 14 (1942) 390. [4] G. Schwarz, Z. Phys. 145 (1956) 563. [5] A. Trusov, V.V. Vojtylov, Electrooptics and Conductometry of Polydisperse Systems, CRC Press, New York, 1993. [6 ] H. Watanabe, A. Morita, Adv. Chem. Phys. 56 (1984) 255. [7] N.A. Tolstoi, A.A. Spartakov, Electrooptika i Magnitooptika Dispersnyh Sistem (Electrooptics and Magnetooptics of Disperse Systems), St Petersburg University, 1996 (in Russian). [8] V.V. Voitylov, A.A. Spartakov, N.A. Tolstoi, A.A. Trusov, Colloid J. 59 (1987) N2 [ Translated from Kolloidn. Zh. 59 (1987) 169]. [9] V. Voitylov, T.Yu. Zernova, A.A. Trusov, Colloid J. 56 (1994) 411 [ Translated from Kolloidn. Zh. 56 (1994) 481]. [10] A. Tikhonov, V.Ya. Arsenin, Metody Reshenia Nekorrektnyh Zadach (Techniques of Solving Ill-posed Problems), Nauka, Moscow, 1979 (in Russian).
Polydispersity of macromolecular solutions and colloids in electro-optics L.K. Babadzanjanz, M.L. Bregman, A.A. Trusov, V.V. Vojtylov * Faculty of Applied Mathematics and Processes of Management, Faculty of Physics, St Petersburg State University, Ulianovskaia 1, Petrodvoretz, St Petersburg 198904, Russia
Abstract This article analyzes the problems of polydispersity in electro-optics. It examines the application of the regularisation technique in order to determine distribution functions of colloidal particles and macromolecules on electric dipole moments, on polarisability anisotropy values, and on rotary friction coefficients. It also includes two additions to the regularisation technique. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Conductivity anisotropy; Dipole moment; Distribution function; Electro-optical methods; Polarisability
where
1. Introduction In what follows we consider methods of investigating the particles of liquid systems. These particles are macromolecules in solutions and colloid particles in disperse systems. The characteristics of these particles are the anisotropy of polarisability c, the permanent dipole moment m and the rotary diffusion coefficient D. These characteristics can be determined by well-known electro-optical methods [1], if the system in question is monodisperse. These include birefringence, dichroism, anisotropy of conductivity and permittivity [2,3]. Mainly, we will consider the polydisperse systems, but we will begin with monodisperse ones. The effects we discuss can be determined for monodisperse systems by the following equation: A=A W max * Corresponding author. Fax: +7 812 114 3387; e-mail: [email protected]
(1)
W=
P
p 3 cos2 h−1 0
2
F(h) sin h dh
(2)
The quantity W is a factor of particle orientation, h is the angle between the field direction and the symmetry axis of the particle or macromolecule and F(h) is the angular distribution function. The function F(h) and the factor W depend on the kind of electric field applied to the system, the strength E of the field, the frequency v, the time t and the particle characteristics c, m, D. The quantity A is the magnitude of electro-orientational effect. Electro-orientational effects consist of: birefringence, dichroism, conductivity anisotropy, and permittivity anisotropy. The quantity A is equal max to A if the orientation of the particles is saturated. To study polydisperse systems we will also determine the distribution function of c, m and D. The number distribution functions show the dependences between the values c, m and D. These
0927-7757/99/$ – see front matter © 1999 Elsevier Science B.V. All rights reserved. PII S0 9 2 7 -7 7 5 7 ( 9 8 ) 0 0 60 0 - 1
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L.K. Babadzanjanz et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 148 (1999) 29–34
dependences may be used when studying such complicated systems as solutions of DNA, polypeptides and suspensions of biological cells. To simplify the problem, the angular distribution function F(h) will depend only on the characteristic m, c or D. The distribution function of the maximum effect A on parameters of particles of polydisperse systems can be determined as follows: ∂A ∂A ∂A f (c)¬ max , f ( m)¬ max , f (D)¬ max 1 2 3 ∂c ∂m ∂D (3) The general equation for these functions is:
P
A(y)=
b
K(y, j) f (j) dj
where c˜ ¬cE2 /2kT, k is the Boltzmann coefficient, 0 and T the absolute temperature. 1.2. Second example: the case of m Here we consider the distribution of permanent dipole moments. There are two subcases represented by different formulae. (a) E=E , y=E, j=m 0 1 3 coth( m˜ ) K(y, j)=W ( m˜ )¬1+ − (7) 2a m˜ 2 m˜ m˜ ¬
mE kT
(4) (b) V&D
a
Here: y=E,
E2 2
K(y, j)=W ( m˜ , c˜ )¬2 2b
1
, v, t; j=c, m, D,
D
f (j)=f (c), f ( m), f (D) (5) 1 2 3 The main problem of this work is that of solving the integral Eq. (4). The analytic formulae for the kernel K(y, j) depend on the different kinds of electrical fields applied to the system. Here we have to note that in all cases we consider the kernel as a function of one variable, namely y2j, yj, and y/j. Now let us consider some examples of such a kernel [4,5]. 1.1. First example: the case of c Let us consider the kernel of the integral equation for anisotropy polarisability distribution with the applied field being sine-shaped and of high frequency [2,6 ]: E=E sin(Vt), V&D 0 y=E2 /2, j=c 0 Here we have: K(y, j)=W (c˜ ) 1 3 1 ¬ x2 ec˜ x2 dx 2 0
P
P
c 0
×[u(x)+u(1−x2)] u(x)¬x3 ec˜ x2
C
P
dx 앀1−x2
1/앀2 1−2y2 0
앀1−y2
A
× coth(c˜ m˜ y)+coth c˜ m˜ 앀1−y2
BD
dy
(8)
(a) is the subcase where the permanent dipole moment is much bigger than the induced one. This happens if the system consists of polar macromolecules or particles [7]. (b) is the subcase when the field direction changes with high frequency V within the limits of a right angle [8]. In many situations the dependence of the kernel on c˜ is negligible because values of the function W of 2b high polarisability are close to values of the function W of low polarisability. 2b 1.3. Third example: the case of D
NP
1 0
ec˜ x2 dx−
1 2
(6)
The third example arises if we consider the rotation friction of the particles in two different subcases corresponding to different kinds of fields applied. (a) High field ( m˜ +c˜ &1). This is a subcase of
31
L.K. Babadzanjanz et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 148 (1999) 29–34
relaxation from the saturated orientation [1,5]:
2.1. Regularisation
y=t, j=D
This method works only for smooth functions. Let us begin with the main equation of the Tikhonov regularisation technique [10] to solving the ill-posed problem (4):
K(y, j)=W (t˜)¬e−t˜ 3 ˜t¬6Dt
(9)
(b) Low field (c˜ %1). This is the subcase of sinefields of high frequency with sine-shaped amplitude modulation of low frequency: E=E sin(Vt) sin(vt) 0 V&D, v∏V (10)
y=v, j=D
The value A of electro-orientational effect can be determined by the following equation [9]: A(v)=
c˜ 30
− v ˜¬
C
1+ 1
1+v ˜2
v ˜ 1+v ˜2
sin(2vt)
D
cos(2vt)
(11)
P
b a
˜ (j , j) f (j) dj −a K 1 1
P
˜ (j, j )¬ K 1
P
d
3D
c yµ[c, d ]
(12)
1 K(y, j)=W (v ˜ )¬ 5 1+v ˜2
(13)
2. Methods and results We will discuss techniques attaining to the following: $ regularisation; $ modification of the kernel; $ equations for derivatives of distribution functions.
D
−f (j) =b(j)
where a is a real constant, which can be calculated by various techniques [5,10]. The regularisation method requires that the solution of Eq. (14) belong to a class of smooth functions. This is natural in case of the physical problems under consideration. The algorithm of the regularisation technique could be briefly described as follows. Step 1. Through the numerical integration technique one obtains the functions:
b(j)¬
v ˜ K(y, j)=W (v ˜ )¬ 4 1+v ˜2
dj2
(14)
v
Sine-shaped and cosine-shaped terms can be measured separately in experiments and give the possibility of determining the following kernels:
C
d2f (j)
d c
K(y, j)K(y, j ) dy 1
(15)
A(y)K(y, j) dy (16)
Step 2. Eq. (14) can be reduced to a system of linear algebraic equations. By solving this system, one finds a numerical solution to the integral Eq. (4). The advantage of this technique is that it is not sensitive to the noise–signal ratio. The chief disadvantage of this technique is that it fails to resolve problems under consideration if multimodal solutions occur. Let us have a look at the result of the numerical simulation of the unimodal solution for kernel numbered (9). This is the case of relaxation, from the saturate orientation of macromolecules or particles. The results are shown in Fig. 1. As one can see, in case of bimodal function the result is much worse. The results are shown in Fig. 2. The other two techniques we have to discuss have been designed as an addition to regularisation techniques in order to improve the results for multimodal solutions.
32
L.K. Babadzanjanz et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 148 (1999) 29–34
we derive:
P
b A (y)= 1 K (y, j) f (j) dj d d a1 where
(17)
∂mA(y) A (y)¬ d ∂ym
(18)
∂mK(y, j) K (y, j)¬ d ∂ym
(19)
[a , b ]5[a, b] 1 1 Fig. 1. Unimodal distribution function. The solid curve is a model curve, the dashed curve is a solution with a 10% random error of A(y), the dot-dashed curve is a solution with a 20% random error of A(y), the value D corresponds to the maxi0 mum of f(D).
Consider two examples: (a) the kernel is given by Eq. (6); (b) the kernel is given by Eqs. (12) and (13). As one can see in Fig. 3(a–c), the graphics of the derivatives of the kernels are non-negligible in a more narrow range of the argument. If the derivative of A(y) is measured in an experiment then one can solve Eq. (17) using the regularisation technique for two-modal functions. It gives more accurate results because the new kernel is narrower than the original one. 2.3. Equations for derivative distribution functions Here we assume that the kernel is a function of one argument yj. Thus we consider the equation:
P
b
A(y)=
Fig. 2. Bimodal distribution function. The solid curve is a model curve, the dashed curve is a solution calculated by the regularisation technique, the value D corresponds to the first maximum 0 of f(D).
K(yj) f (j) dj
(20)
a
where K(y, j)=K(yj) Besides, we assume: f (m)(a)=f (m)(b)=0, m=0, 1, 2, …
(21)
2.2. Modification of the kernel The kernel modification method is based on the differentiation of the original Eq. (4). In such a way the kernel reduces to another one which changes in a more narrow gap [a , b ] than [a, b]; 1 1
Combining the procedures of differentiation and integration by parts, we derive the formulae: A (y)= m
P
b a
K(yj)jm f(m)(j) dj
(22)
L.K. Babadzanjanz et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 148 (1999) 29–34
33
where A (y)=A(y) 0
(23)
=(1−m)A −y∂A (y)/∂y, m=1, 2, … m+1 m m Eq. (22) has the same form as Eq. (4), and it is an integral equation on the product of jm and f (m)(j)¬∂m f/∂ym. If m=0 then Eq. (22) is the same as the original Eq. (20). For m greater than 0 it becomes the integral equation with respect to m multiplied by the derivative of order m with the kernel being the same. The scheme of the technique under consideration for m=1 is as follows. Step 1. We find the data: A
A(y ), A(y ), …, A(y ) (24) 1 2 k A (y ), A (y ), …, A (y ) (25) d 1 d 2 d k from the experiment, and then we get analytical approximations for A of y and its derivative. Step 2. Using the regularisation technique we get a numerical solution of the equations:
P
A (y)= 0
b
K(yj) f (j) dj
(26)
a
A (y)=A (y)−y 1 0
P
∂A (y) b 0 = K(yj)jf ∞(j) dj ∂y a (27)
namely the data: f (j ), f (j ), …, f (j ) (28) 1 2 n j f ∞(j ), j f ∞(j ), …, j f ∞(j ) (29) 1 1 2 2 n n Step 3. Using the results of Eqs. (28) and (29) we construct an analytical formula for f(j). For example, at Steps 1 and 3 we can use splines constructed by a least-squares approximation method. Fig. 3. Graphs of kernels and their derivatives. (a) The solid curve is function W (c˜ ), the dashed curve is function dW /dc˜ , 1 1 the dot-dashed curve is d2W /dc˜ 2. (b) The solid curve is func1 tion W (v ˜ ), the dashed curve is function dW /dv ˜ . (c) The solid 4 4 curve is function W (v ˜ ), the dashed curve is function dW /dv ˜. 5 5
3. Conclusion So, it is stated (Section 2.1) that the Tikhonov regularisation technique for solving the ill-posed
34
L.K. Babadzanjanz et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 148 (1999) 29–34
problems of polydispersity in electro-optics is effective if the solution (distribution function) occurs as a one-modal function and fails if it has two or more modes. To improve this method, one could use the two techniques suggested above (Sections 2.2 and 2.3). The first technique can be used to change the original ill-posed problem to a different ill-posed problem. Applying the regularisation technique to the latter problem, one gets a more accurate result because the domain of the new kernel is narrower than the original one. The second technique gives an opportunity to find the derivative of the solution of the original integral equation and, in particular, to recognize how many modes the solution has. Further modeling experiments are to be carried out in order to investigate these techniques more thoroughly.
Acknowledgment The authors thank the Russian Fond for Fundamental Investigations for financial support
(Grant RFFI 98-03-32713a and Grant RFFI 96-03-34310a).
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