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from Turbidimetric Measurements Combining Regularization and Generalized Cross-Validation Techniques Guillermo E . Eliçabe and Luis H . Garciá-Rubio Chemical Engineering Department, College of Engineering, University of South Florida, Tampa, FL 33620

This chapter reports the recovery (deconvolution) of particle size distributions (PSDs) from turbidimetric measurements using a regularization technique (RT). Regularization techniques require the selection of a constraining parameter known as the regularization parameter. In this work the regularization parameter was calculated by using the generalized cross-validation (GCV) technique. The use of these complimentary techniques (RT and GCV) is demonstrated through the simulated recovery of PSDs of polystyrene latices. Unimodal and bimodal PSDs of varying breadth and mean particle diameters were investigated. The results demonstrate that the combination of these techniques yields adequate recoveries of the PSDs in almost every case. The cases where the techniques fail are identified, and strategies for subsequent recovery are discussed.

^VHEN

A S U S P E N S I O N O F S P H E R I C A L P A R T I C L E S is illuminated with light of different wavelengths, the resulting optical spectral extinction (turbidity) contains information that, in principle, can be used to estimate the particle size distribution (PSD) of the suspended particles. The recovery (deconvolution) of the P S D from turbidity measurements falls within the category of "inverse problems" to which several techniques have been applied with varying degrees of success (1-5). Recently (6), a regularization technique

0065-2393/90/0227-0083$06.25/0 © 1990 American Chemical Society

In Polymer Characterization; Craver, C., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1990.

84

POLYMER CHARACTERIZATION

was successfully a p p l i e d to the estimation of the P S D s of p o l y s t y r e n e latices. R e g u l a r i z a t i o n techniques r e q u i r e the selection of a c o n s t r a i n i n g p a r a m e t e r 7, k n o w n as the regularization parameter. T h e selection of the r e g u l a r i z a t i o n p a r a m e t e r is c r i t i c a l for the adequate r e c o v e r y of the P S D (6).

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In this c h a p t e r some o f the available methods for the estimation of y are b r i e f l y i n t r o d u c e d . P a r t i c u l a r emphasis has b e e n p l a c e d o n the genera l i z e d cross-validation ( G C V ) t e c h n i q u e . T h i s t e c h n i q u e , i n o u r a p p l i c a t i o n , appears to b e the most robust a m o n g the techniques available. I n the f o l l o w i n g section, the equations that relate the particle size a n d the t u r b i d i t y are s h o w n , a n d a discrete m o d e l for these equations is d e s c r i b e d i n d e t a i l . T h e n , the r e g u l a r i z e d solution o f the discrete m o d e l p r e v i o u s l y d e v e l o p e d is i n t r o d u c e d . A discussion about some of the techniques available for e s t i m a t i n g the r e g u l a r i z a t i o n p a r a m e t e r is g i v e n . T h e G C V t e c h n i q u e is t h e n r e v i s i t e d . F i n a l l y , the results of s i m u l a t e d examples are s h o w n . I n a l l the examples, the r e g u l a r i z e d solution is u s e d along w i t h the G C V t e c h n i q u e to estimate a b r o a d range o f P S D s of p o l y s t y r e n e latices.

Absorption and Light Scattering of Spherical Suspended Particles T h e loss of i n t e n s i t y e x p e r i e n c e d b y a b e a m of electromagnetic r a d i a t i o n i n passing t h r o u g h a sample of s u s p e n d e d particles, r e c o r d e d as a f u n c t i o n of the w a v e l e n g t h of the i n c i d e n t radiation, is k n o w n as the t u r b i d i t y s p e c t r u m . T h e t u r b i d i t y (T) is r e l a t e d to the intensities at two points separated a distance Iby

(1) w h e r e 1 is the i n t e n s i t y at the p o i n t w h e r e the electromagnetic r a d i a t i o n enters the sample, a n d it coincides w i t h the i n t e n s i t y of the source; I is the intensity at the p o i n t w h e r e the electromagnetic radiation leaves the s a m p l e , a n d it coincides w i t h the i n t e n s i t y at the detector. F o r a suspension of monodisperse isotropic s p h e r i c a l particles, the t u r b i d i t y can be r e l a t e d to the w a v e l e n g t h o f the i n c i d e n t radiation (X ), the particle d i a m e t e r (D), a n d the optical p r o p e r t i e s of the suspension t h r o u g h M i e theory (7): 0

0

T ( X , D ) = N ^ D'QMKol 0

p

*i(Ao), % ( U X , D] 0

(2)

w h e r e N is the total n u m b e r o f particles p e r u n i t v o l u m e i n the sample a n d Q is the e x t i n c t i o n efficiency. Q is a f u n c t i o n of (1) the real a n d i m a g i n a r y parts of the particle refractive i n d e x (n a n d k respectively); (2) the refractive i n d e x of the suspension m e d i u m , n ; (3) the w a v e l e n g t h of the i n c i d e n t radiation i n vacuo; a n d (4) the d i a m e t e r of the s p h e r i c a l particles. T h e r e p

ext

e x t

1

h

2


6.

ELIQABE

fractive

Turbidimetric Measurements of PSD

or G A R C I A - R U B I O

85

indexes are, i n general, functions of the w a v e l e n g t h . E q u a t i o n 2 can

be r e a d i l y expressed i n terms of the particle concentration (i.e., C is the w e i g h t of particles p e r u n i t v o l u m e of suspension): 3C T(X , D ) = — g (\o, D ) 0

(3)

ext


w h e r e p is the d e n s i t y of the p a r t i c l e . ( F o r s i m p l i c i t y , the refractive indexes have b e e n o m i t t e d from the a r g u m e n t of Q

from e q 3 onward).

ext

I f the sample is a m i x t u r e w i t h a d i s t r i b u t i o n of particle diameters, a n d the P S D can b e r e p r e s e n t e d b y a differential d i s t r i b u t i o n , the t u r b i d i t y can be r e w r i t t e n as 00

D ) D / ' ( D ) dD 2

(6)

0 where now 00

f'ip) dD = 1

/

(7)

S i m i l a r l y , the t u r b i d i t y of a polydisperse suspension, i n terms of c o n c e n t r a t i o n C , can be w r i t t e n as

| g (X , D)D*f'(D) ext

0

dD

_0 2p

(8)

oo

| f'(D)D

3

dD


86


By defining:

K(Xo, D) = j

Qjfo,

D)D

(9)

2

eq 4 can be readily identified as a Fredholm integral equation of the first kind, in which K(X ,D) is the corresponding kernel. The numerical solution 0


to any of these equations (eq 4, 6, or 8) must be based on an appropriate discrete model. The solution to such a model will result in estimates of the number of particles and of the shape of the P S D . If the integrand in eq 4 is discretized into n-l wavelength k

intervals, the integral can be approximated at a given

with a sum,

0i

2

T, «

a.fj

(10)

j=i where T = T(\ J) andfj = f(D ). The details of the discretization procedure and the resulting coefficients a are given in the Appendix. (

}

0

0

If the turbidity is evaluated at m wavelengths \ , 4 can be written in matrix form, 0i

i = 1, . . . , m, eq

— Af

T

(11)

where T

=

[T,T

T F

(12)

m

2

A = {aj f=[/i/

(13) fnY

2

(14)

T indicates the transpose. Equation 11 can be written as an equality if the quadrature error € introduced in the discretization is considered, T

= Af + e

c

(15)

c

Finally, with the addition of the measurement error e , the discrete equation m

for the representation of the experimental values of T (i. e., T J , can be written as T

M

= Af + € + € c

m

= Af + €


(16)

6.

EugABE

87


& GARCIA-RUBIO

Particle Size Distribution from Turbidity Measurements The

Solution of Equation 16 Using Regularization Techniques.

discrete m o d e l d e v e l o p e d i n the previous section (i.e., e q 16) transforms the p r o b l e m of o b t a i n i n g the P S D from t u r b i d i t y measurements into a l i n e a r algebraic p r o b l e m , w h e r e n points of the P S D can be estimated from m t u r b i d i t y measurements (m has to be greater than or e q u a l to n). I f m = n, estimates of the P S D (f ) can, i n p r i n c i p l e , be o b t a i n e d b y the d i r e c t i n v e r s i o n of e q 16: Downloaded by UCSF LIB CKM RSCS MGMT on November 19, 2014 | http://pubs.acs.org Publication Date: May 5, 1990 | doi: 10.1021/ba-1990-0227.ch006

d

(17)

h = A" ^ 1

A l t e r n a t i v e l y , i f m > n , the least-squares solution of an overspecified system of l i n e a r equations yields f

ls

=

(18)

(A A)-Wr r

m

A l t h o u g h these solutions appear to be straightforward, it is w e l l d o c u m e n t e d i n the l i t e r a t u r e (8-12) that small ^errors (i.e., quadrature a n d exp e r i m e n t a l errors) result i n large errors i n f or f . T h e amplification of the errors occurs i n d e p e n d e n t l y of the fact that the inverses of A a n d ( A A ) can be calculated exactly, a n d it is a d i r e c t consequence of the near singularity of the m a t r i x A (if ra = n), or m o r e generally (if m > n) of its near i n c o m p l e t e rank. T h i s b e h a v i o r can be e x p l a i n e d b y the near l i n e a r d e p e n d e n c e b e t w e e n the functions K(\ ,D) from w h i c h the matrix A was o b t a i n e d . I n spite of the fact that these functions d e p e n d o n the optical properties of the system u n d e r study, the p r e s e l e c t e d range of wavelengths a n d diameters, a n d the n u m b e r of points that it is d e s i r e d to recover, a certain a m o u n t of collinearity b e t w e e n some of the functions w i l l be always present. A d d i n g the fact that at least a s m a l l e x p e r i m e n t a l e r r o r is also always present, it is possible to state that eqs 17 a n d 18 cannot give a solution to the p r o b l e m u n d e r study. d

ls

T

0i

H o w e v e r , b y c o n s t r a i n i n g the least-squares solution b y means of a p e n alty f u n c t i o n , approximate useful solutions can be obtained. T h i s step can be a c h i e v e d b y u s i n g a l l of the p r i o r information available r e g a r d i n g the P S D (i.e., the " t r u e " f vector). F o r example, it is k n o w n that the values of f m u s t b e positive or zero, that there is an u p p e r a n d a l o w e r b o u n d o n the particle diameters, a n d that a c e r t a i n amount of correlation exists b e t w e e n successive points o n the d i s t r i b u t i o n . Because e q 18 is the solution to the least-squares p r o b l e m , t h e n m i n |Af -

T | M

2

f

w h e r e | • | indicates the m o d u l u s , a n d f

ls

has b e e n r e p l a c e d b y f.


(19)

88


P r i o r i n f o r m a t i o n can b e i n t r o d u c e d b y a u g m e n t i n g e q 19 w i t h (8-12)


m i n [|Af -

Tra

| + yqfo]

(20)

2

w h e r e q(t) is a scalar function that measures the correlation or smoothness o f f , a n d 7 is a nonnegative parameter that can b e v a r i e d to emphasize m o r e or less one of the terms of the objective functional g i v e n b y e q 20. I f 7 is set to 0, e q 20 reduces to e q 19, a solution that generally exhibits large oscillations. O n the other h a n d , w h e n 7 —> the m i n i m i z a t i o n leads to a perfectly smooth solution j u d g e d b y the measure of q(f) b u t totally i n d e p e n d e n t of the T values a n d , therefore, useless. C l e a r l y , i n t e r m e d i a t e values of 7 w i l l p r o d u c e acceptable solutions to the o r i g i n a l p r o b l e m (i.e., e q 16) and those solutions w i l l have the smoothness or correlation characteristics i m p o s e d b y the t e r m q(f) i n the functional. A l s o , it has b e e n d e m o n s t r a t e d that for b o u n d e d (f f), there exists a value of 7 > 0 such that (13) 0 0

M

T

E[(f

-

fV({

-

()] < E[(f

-

u

f)F(f

ls

-

f)]

(21)

w h e r e E[] indicates expected value. I n other w o r d s , the e r r o r i n the estimation of f associated w i t h the solution of e q 20 w i l l be s m a l l e r than that associated w i t h the solution of e q 19. H o w e v e r , an appropriate form must be selected for the function q(f) and an adequate value for the parameter 7. Several functions can b e chosen to establish the d e s i r e d correlation l e v e l or the smoothness of f. A n i n t e r e s t i n g class of functions can be f o r m u l a t e d b y u s i n g a q u a d r a t i c form of the vector f because t h e y y i e l d an analytical solution to the m i n i m i z a t i o n p r o b l e m of e q 20. F o r example, i f q(£) = f f , e q 20 can be r e a d i l y i d e n t i f i e d w i t h the w e l l - k n o w n r i d g e regression (14). A m o r e i n t e r e s t i n g example i n w h i c h r

q(f) =

FK Kf T

with

K

0

0

•

•

•

•

1

-1

0

•

•

•

0 0

0

1

-1

0

•

•

0

=

(22) 0 •

1 0


6.

ELigABE

89


& GARCIA-RUBIO

gives the following q(l)

q(h

(23)

= i(fj-fj-i)

2

j =2 A


w h i c h is a t y p i c a l measure o f smoothness. A n o t h e r less restrictive 17(f) is given b y the sum o f the squares o f the second differences

= " f (2fj - fj-t

q$

(24)

- f f j+l

j=2

In this case the matrix H = K K is given b y r

-2 5 -4

1

-2 1

1

-4 6

• • 0 • •

0 0 0

• 0• • •

0 1 -4

1

-4

1

1

6 -4

0

1

-4 5 -2

0 0 0 (25) 1

-2 1

It w i l l b e shown later that u n i m o d a l a n d b i m o d a l latex distributions can be readily analyzed b y u s i n g this last quadratic form w i t h a slight m o d ification that constrains t h e values o f / i a n d / „ to b e 0, thus i n c o r p o r a t i n g into t h e solution additional p r i o r k n o w l e d g e that h a d b e e n i m p o s e d d u r i n g the d e r i v a t i o n of the discrete m o d e l . T h i s last constraint can b e i m p l e m e n t e d b y s u m m i n g |3 , w i t h 0 > > 1, to the (1,1) a n d (n,n) elements o f the H matrix shown i n e q 25. I n this form t h e final quadratic form of the q(f) function for the examples o f the following sections w i l l b e 2

q(f) =

TO

2

+ A ) + I 2

- f f

(2/, -

j+1

(26)

j=2

H a v i n g a r r i v e d at an explicit expression for H , w e can show that the solution to t h e constrained p r o b l e m o f e q 20 is given b y ( I I ) : f = (A A + 7 H ) " A T r

1

R

M

(27)

T h e value o f f obtained w i t h e q 27 w i l l b e c a l l e d the r e g u l a r i z e d solution of e q 16. I f the matrix H o f e q 2 5 is used w i t h t h e modification that p e r m i t s the constraint f = / „ = 0, (A A + yH) is a positive definite s y m m e t r i c matrix. T h u s , efficient algorithms c a n b e u s e d to perform t h e inverse. x

r


90



Selection of y. T h e r e g u l a r i z e d solution o f e q 27 requires t h e selection o f the regularization p a r a m e t e r 7. T h e existing m e t h o d s for selecting 7 can b e r o u g h l y d i v i d e d i n two: those s t e m m i n g from applications i n physics a n d e n g i n e e r i n g , a n d those d e v e l o p e d i n statistics. A m o n g the first, T w o m e y ' s analysis o f information content ( I I ) has b e e n a p p l i e d mostly i n atmospheric sciences. T h e i d e a o f T w o m e y ' s m e t h o d is to detect t h e n u m b e r o f i n d e p e n d e n t pieces o f information available i n a set of e x p e r i m e n t a l measurements. T h i s analysis leads to a regularization p a rameter 7 that also d e p e n d s o n an estimation o f the root-mean-square value of the m e a s u r e m e n t noise. U s i n g a c o m p l e t e l y different approach, P r o v e n c h e r (15) p r o p o s e d a m e t h o d for selecting 7 i n t h e p r o b l e m o f i n v e r t i n g the F r e d h o l m i n t e g r a l equation that arises i n the d e t e r m i n a t i o n o f m o l e c u l a r w e i g h t distributions ( M W D ) o f p o l y m e r s u s i n g p h o t o n correlation spectroscopy. Proveneher's m e t h o d is analogous to t h e standard p r o c e d u r e o f c o n structing confidence regions for t h e sought solution. A l t h o u g h t h e m e t h o d is rather a r b i t r a r y , the results o b t a i n e d for the estimation o f the M W D w e r e satisfactory. T h e methods for the selection o f 7 based o n statistics theory w e r e d e v e l o p e d for the so-called ridge regression (RR). A s stated p r e v i o u s l y , R R is a special case o f e q 2 7 i n w h i c h H is t h e i d e n t i t y matrix (I). B y d e f i n i n g X = AK"

(28)

1

and f' = K f

(29)

then f

= (X X T

+ yl)~ X 7 l

T

m

(30)

Therefore, the r e g u l a r i z e d solution o f e q 27 can b e seen as a R R , a n d t h e methods specifically d e v e l o p e d for estimating 7 i n e q 3 0 c a n b e d i r e c t l y a p p l i e d to e q 2 7 . T h e statistical methods for t h e estimation o f 7 m a y b e d i v i d e d into two: those that use a p r i o r i i n f o r m a t i o n , a n d those that use o n l y the m e a s u r e d data. A m o n g the first, the m e t h o d o f H o e r l et a l . (14,16) a n d the c l o s e d f o r m solution for t h e iterative m e t h o d d e s c r i b e d i n ref. 16 g i v e n b y H e m m e r l e (17) c a n b e c i t e d . I n these cases a n estimation o f the variance o f the noise (a ) a n d an i n i t i a l estimate o f the solution are n e e d e d . A n o t h e r m e t h o d , w h i c h uses o n l y a n estimate o f a , is k n o w n as t h e "range r i s k " estimate, a n d it is briefly o u t l i n e d i n ref. 18. A c c o r d i n g to G o l u b et a l . (18), t h e o n l y m e t h o d s available for estimating 7 from t h e data are m a x i m u m l i k e l i h o o d , o r d i n a r y cross validation ( O C V ) , a n d generalized cross validation ( G C V ) . 2

2


6.

EugABE

& GARCIA-RUBIO


91

T h e s e t h r e e methods d o not r e q u i r e any a p r i o r i i n f o r m a t i o n , a n d therefore they c a n b e u s e d to c o m p l e t e l y automatize t h e P S D e s t i m a t i o n process. S i m u l a t e d studies (18, 19) have s h o w n that t h e G C V t e c h n i q u e is t h e most r e l i a b l e a n d the best t h e o r e t i c a l l y f o u n d e d a m o n g those u s i n g o n l y t h e m e a s u r e d data. T h i s assertion justifies, i n p r i n c i p l e , its selection as a m e t h o d for e s t i m a t i n g 7 i n t h e context o f P S D e s t i m a t i o n from t u r b i d i m e t r i c data. T h e Generalized Cross-Validation Technique.

T h e G C V tech-


n i q u e is a rotation-invariant v e r s i o n o f O C V . T h e O C V t e c h n i q u e m a y b e d e r i v e d as follows: define f (y) as t h e e s t i m a t i o n o f f ,{k)

w i t h t h e kth value o f T

[ X f ' % ^ (the kth c o m p o n e n t o f t h e [Xf (y)]

vector) s h o u l d b e a good

,{k)

predictor of T

MFC

= K f u s i n g e q 30

o m i t t e d . T h e a r g u m e n t is that i f 7 is adequate, t h e n

M

(the kth c o m p o n e n t o f T J . I n o r d e r to obtain good p r e d i c t o r s

for a l l t h e measurements

1, . . . , m), y s h o u l d b e chosen as t h e

(k =

minimizer of

m

= -

2

flxH-Y)]*

-

v>

(3D

2

This function can be expressed as (18):

P(y) = - \B(y)[I m

Z( )]x | 7

m

(32)

2

w h e r e B(y) is a diagonal matrix w i t h entries {1/[1 - 2^(7)]} w h e r e t h e e l e m e n t Zjj is the jj e n t r y o f Z(y) = X(X X T

+ 7l)" X 1

r

A l t h o u g h t h e i d e a d e v e l o p e d as j u s t d e s c r i b e d seems to b e a p p e a l i n g , G o l u b et a l . (18) p o i n t e d o u t that this m e t h o d fails w h e n the m a t r i x Z(y) is diagonal because P(y) does n o t have a u n i q u e m i n i m i z e r . T h i s b e h a v i o r indicates that the O C V is not e x p e c t e d to p e r f o r m successfully i n t h e near diagonal case e i t h e r . T o c i r c u m v e n t this difficulty, t h e G C V t e c h n i q u e was i n t r o d u c e d as a rotation-invariant f o r m of O C V (18). T h e G C V function o f 7 can b e d e f i n e d as the O C V f u n c t i o n (eq 32) a p p l i e d to t h e f o l l o w i n g transformed m o d e l T = WL7 T R

W

= WDVT

+ WU e T

= Xf' + WU e T

(33)

w h e r e U a n d V are t h e result o f the singular value d e c o m p o s i t i o n o f X X =

UDV

T


(34)

92


and W is a c o m p l e x matrix w h o s e elements are

Wij = 77= exp ( ? ^ 2 * )

j

k = 1, 2, ••• m

y

(35)

where V i ~ = -1.


T h e r e f o r e , u s i n g the transformed m o d e l i n e q 3 2 results i n

m

= -

imu

-

Z(-/)]T|

(36)

2

m

U s i n g t h e fact that as a result o f the transformation, Z(y) is a c i r c u l a n t matrix a n d h e n c e constant d o w n t h e diagonals, the last equation c a n b e expressed as

{trace [I -

Z(y)]}

2

It c a n also b e s h o w n that 12 2

V(-y) =

w h e r e z = [z , x

m

j

{trace [J -

. . . , z] m

——— = Z(y)]}

2

= C/ T T

M

m

-r-

rr

n

I " 7 >. \«tl X, + y

h

m

-

n

\

2

(38)

J

a n d X, (i = 1, . . . , n) are t h e e i g e n v a l u e s

of(X X). T

Recovery of the Particle Size Distribution for Polystyrene Latices I n this section, eqs 27 a n d 38 w i l l b e u s e d to estimate the P S D s of p o l y s t y r e n e latices. U s i n g t h e measurements a n d t h e m o d e l , e q 38 w i l l b e m i n i m i z e d w i t h respect to y. T h e value o f y that m i n i m i z e s e q 38 w i l l b e t h e n u s e d i n e q 27 to estimate t h e P S D s . T h e t u r b i d i t y spectra w e r e s i m u l a t e d b y u s i n g the results d e s c r i b e d u n d e r " A b s o r p t i o n a n d L i g h t Scattering o f S p h e r i c a l S u s p e n d e d P a r t i c l e s " . F o r t h e s i m u l a t e d e x p e r i m e n t s , t h e refractive i n d e x of water was calculated from (20)

(39)


6.

EugABE

93


& GARCIA-RUBIO

w i t h X g i v e n i n nanometers. T h e real a n d the imaginary parts of the c o m p l e x refractive i n d e x for polystyrene w e r e obtained from the data of Inagaki et al. (21). F i g u r e 1 shows the optical properties of polystyrene a n d water as functions of the w a v e l e n g t h . F i g u r e 2 shows the distributions analyzed. A b r o a d range of possibilities is b e i n g c o n s i d e r e d , i n c l u d i n g b i m o d a l a n d v e r y n a r r o w distributions. T h e mathematical expression for those distributions is


0

«

=4 ^

H S , t

ST?. ° '''] 1

8

(40>

where N

p

= 8.49 X 10 (particles per cubic centimeter)

(41)

6

and

W l

0

N g

. =

N

l

exp r - f r C P - P J - h P J ' l

I V2rro-j(D -

D) s

[_

2of

(42)

J

Table I shows the values for the l e a d i n g parameters that characterize the distributions. T h e s i m u l a t e d t u r b i d i t y spectra w e r e calculated b y u s i n g a 51-point discretization of the distributions generated w i t h e q 40 a n d the parameters s h o w n i n T a b l e I. T h e range of wavelengths was chosen b e t w e e n 200 a n d

A -yn LJ 200 u

o

1 375

1 550 XJnrn]

1 725

Figure 1. Optical parameters of polystyrene (n

»—In 900

lf

k j and water (n ).


2

94


Dmin


f(D)

A

50

Dmax 325

B

50

1300

C

50

3950

D

50

2600

E

50

U50

F

50

2550

2050 2325

G

i

AE

7 50

/

> 1025

2000 D[nm]

2975

3950

Figure 2. PSDs used in the simulated experiments. 900 n m w i t h a r e s o l u t i o n o f 1 n m that results i n a value o f m = 701. A 3 % m a x i m u m value r a n d o m noise (relative to the m a x i m u m t u r b i d i t y value) was a d d e d to the s i m u l a t e d spectra to represent extreme m e a s u r e m e n t conditions (in t h e i n s t r u m e n t , noise is less than 0.01 absorption units). T h e use o f an exaggerated noise l e v e l demonstrates that t h e t e c h n i q u e w o u l d b e robust for t y p i c a l m e a s u r e m e n t errors. T h e s i m u l a t e d spectra w i t h t h e a d d e d noise constitute t h e e x p e r i m e n t a l data. T h e n u m b e r o f r e c o v e r e d points o n t h e d i s t r i b u t i o n s was n = 51 i n a l l cases, a n d the ranges of diameters v a r i e d a c c o r d i n g to the case b e i n g a n a l y z e d (see F i g u r e 2). T h e value o f (3 was chosen as 1000. T o d r a w t h e most general conclusions, a M o n t e C a r l o type e x p e r i m e n t was c a r r i e d out. E a c h spc t r u m was r e p l i c a t e d five times, k e e p i n g t h e statistics of the noise constant. T h e objective function ryfy) was d e f i n e d as y

2

i=l

(43)

i/

2

T h e value o f 7 that m i n i m i z e s e q 43 w o u l d give the best solution i n t h e context o f t h e r e g u l a r i z a t i o n t e c h n i q u e u s e d i n this w o r k . U n f o r t u n a t e l y , this objective function cannot b e evaluated i n a r e a l situation because i t depends o n t h e u n k n o w n value o f / . H o w e v e r , i t p e r m i t s us to e x a m i n e , i n a s i m u l a t e d e x p e r i m e n t , the performance of e q 38 as estimator o f 7.


6.

EugABE

& GARCIA-RUBIO


95

Table I. Parameters That Characterize Particle Size Distributions Used in the Simulated Experiments


Case

cr

c,

— — — —

c

1

0

0

1

0

0

1

0

0

1

0

0

1000

0.1

1

2

0

0.2

1500

0.1

2

1

0

0.2

—

—

1

0

2000

D 2 (nm)

Dg2 (nm)

g

A B C

1000

0.65

D

1300

0.3

E F

600

0.2

600

G

175

175

0.2

600

0.3

2

a

2

D (nm) s

N O T E : Symbols are the same as in eqs 4 0 - 4 2 .

F o r each case, i n c l u d i n g replications, eqs 38 a n d 43 w e r e m i n i m i z e d . T h e values o f 7 that m i n i m i z e those equations, 7 and 7 , respectively, are s h o w n i n T a b l e I I . T h e last t w o c o l u m n s o f that table show the v a l u e o f the o p t i m a l objective f u n c t i o n (eq 43) evaluated at 7 a n d 7 cv- T h e s e data enable us to j u d g e t h e accuracy o f the solution o b t a i n e d w i t h t h e value o f 7 that m i n i m i z e s t h e G C V objective f u n c t i o n (eq 38) w i t h respect to that obtained using 7 . G C V

opt

opt

G

opt

F i g u r e s 3 a n d 4 show eqs 38 a n d 43 as functions o f 7 i n l o g - l o g plots for cases B a n d F , respectively. T h e values o f 7 range from the 3 7 t h to t h e 51st eigenvalue o f the c o r r e s p o n d i n g ( A A ) matrix i n b o t h cases. F o r each case t w o replications are p l o t t e d to show closeness along t h e 7 axis. T w o replications o f the same case s h o u l d give values o f 7 close together for e q 43, a n d thus indicate t h e v a l i d i t y o f t h e 7 values p r o v i d e d b y e q 38 relative to the o p t i m a l values. I n these figures the scales o f the ordinate axis are different for each plot to clearly c o m p a r e t h e locations o f the m i n i m a attained for each function. (The use o f the same scale does not give any additional i n f o r m a t i o n a n d prevents a clear comparison). T

opt

G C V

A s can b e seen i n Table I I , the results are v e r y good i n almost a l l cases. T h e m e t h o d is able to d i s t i n g u i s h b e t w e e n b i m o d a l a n d u n i m o d a l d i s t r i butions. F o r e x a m p l e , w h e n t h e d i s t r i b u t i o n is b i m o d a l , as i n case F , t h e o p t i m u m 7 values are shifted to the left w i t h respect to the u n i m o d a l cases (e.g., case B) to a l l o w the i n h e r e n t oscillations o f a b i m o d a l d i s t r i b u t i o n (see F i g u r e s 3 a n d 4). I n case A , the values of 7 p r o v i d e d b y e q 38 do not give correct solutions i n any r e p l i c a t i o n . T h e d i s t r i b u t i o n c o r r e s p o n d i n g to this case is b e i n g v e r y n a r r o w a n d has a v e r y small n u m b e r - a v e r a g e p a r t i c l e diameter. To d e t e r m i n e i f the p o o r results are d u e to b o t h characteristics o r i f t h e y d e p e n d o n l y o n one o f t h e m , a s i m u l a t i o n u s i n g t h e same d i s t r i b u t i o n o f case A b u t shifted to t h e large-particle diameters was a n a l y z e d . T h i s s i m u l a t i o n corresponds to case G a n d reveals that, a l t h o u g h for this case the best solutions p r o v i d e d b y the r e g u l a r i z a t i o n t e c h n i q u e are not as good as those for case A , the G C V t e c h n i q u e gives values o f 7 v e r y close to t h e o p t i m a l ones. O n t h e o t h e r h a n d , t h e results o b t a i n e d for case C show that, e v e n


96


Table II. Results for the Five Replications of Each Experiment (A to G) Using Equations 38 and 40 Case

Rep. 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

A

B Downloaded by UCSF LIB CKM RSCS MGMT on November 19, 2014 | http://pubs.acs.org Publication Date: May 5, 1990 | doi: 10.1021/ba-1990-0227.ch006

lopt

a

C

D

E

F

G

1.32 1.32 1.32 2.47 1.61 2.05 6.20 7.83 6.20 9.73 2.50 1.95 1.39 2.50 2.05 9.05 9.05 1.94 3.88 9.27 1.43 1.04 1.43 3.24 1.38 9.31 2.69 2.25 9.31 4.78 1.41 1.41 1.41 1.41 9.54

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

7 GCV

IO IO IO IO IO IO IO IO IO IO 1010ioioIO

15 15 15 15 16 12 12 13 12 12

9 9 9 9

10

1 0

-io

IO ioIO IO IO IO IO IO IO IO IO IO io10 IO 10" 10" IO IO

10 9

10

11 12

12 12 12

11 13 12 12 13

12 11

11 12

7.51 5.86 1.70 1.61 1.32 6.20 6.20 9.73 9.73 6.20 6.93 1.78 1.06 1.39 1.95 1.42 9.05 3.71 1.94 1.94 5.04 4.65 6.06 3.24 4.65 1.33 3.96 5.60 3.96 2.25 1.41 3.94 1.41 1.41 1.41

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

IO IO 10

18 17

-20

IO IO IO IO IO IO IO 1 0

16 15 12 12 12 12 12

-io

20-10 20-io

ioioio-

9 9 9

20-io

ioioioIO IO IO IO IO IO IO IO IO IO IO ioioioIO

9 9 9 13 12 12 12 12

11 12 12 12

12

11 9 11

11

11

Tf (lopt) 0.1817 0.1626 0.2216 0.1665 0.1182 0.0833 0.0575 0.0486 0.0671 0.0602 0.0780 0.0788 0.0770 0.0770 0.0727 0.0621 0.0450 0.0687 0.0412 0.0476 0.1974 0.1196 0.1653 0.1117 0.1960 0.2484 0.1736 0.2062 0.1417 0.1886 0.2909 0.3829 0.2303 0.3049 0.1917

1.6838 0.4784 8.3119 0.5952 0.2014 0.0919 0.0575 0.0805 0.0725 0.0681 0.0955 0.1155 0.0937 0.0785 0.0779 0.0624 0.0450 0.1745 0.0541 0.0593 0.2177 0.1535 0.1924 0.1117 0.2062 0.3169 0.1763 0.2174 0.1701 0.2085 0.2909 0.4909 0.2303 0.3049 0.2049

Replication.

a

t h o u g h t h e c o r r e s p o n d i n g d i s t r i b u t i o n has a h i g h n u m b e r o f s m a l l particles, the results o b t a i n e d w i t h t h e G C V t e c h n i q u e are still good. T h e r e f o r e , t h e p o o r b e h a v i o r i n case A m a y b e a t t r i b u t e d to a v e r y n a r r o w d i s t r i b u t i o n i n the small-diameters zone. T h e r e f o r e , t h e G C V t e c h n i q u e is e x p e c t e d to w o r k p o o r l y w h e n the d i s t r i b u t i o n s are v e r y n a r r o w a n d have a s m a l l n u m b e r average particle d i a m e t e r . A clearer p i c t u r e o f the results c a n b e seen i n F i g u r e s 5 - 1 0 i n w h i c h the estimated P S D s for cases B to G are s h o w n , respectively. I n these figures the t r u e d i s t r i b u t i o n a n d two replications are p l o t t e d for each case. A l t h o u g h


ELIQABE & GARCIA-RUBIO



6.


97

98

POLYMER

A=

EXP B

/// is


;/ /

li

CHARACTERIZATION

true P S D —rep. 3 rep. 5

A A A \

/I

f(D)

it

II It

It li

It

£l

//

\

//! /if y > /

/

«

50

675 D[nm]

1300

Figure 5. Case B. True PSD (—) and estimated PSDs using GCV for replications 3 (—) and 5 (• • •).

• true PSD rep. 1 rep. 5

EXP C

f(D)

50

2000 D[nm]

3950

Figure 6. Case C. True PSD (—) and estimated PSDs using GCV for replications 1 (—) and 5 (•• •).


6.

EugABE &


GARCIA-RUBIO

true P S D

99

EXP D

ffi

rep. 5 rep. 3

,// L


SI

si

f(D)

y'

ii li

, \.

y

/

s

V

- J1

L i J 2600

ii 1325 D[nm]

50

Figure 7. Case D. True PSD (—) and estimated PSDs using GCV for replications 5 (—) and 3 (• • •).

— true P S D — rep. 2 rep. 5

EXP E

'"\ \ if ij i]

a ih 9*

f(D) /

1

\\

A

>o.

.....

•~ /* -

•

50

1

750 D[nm]

1450

Figure 8. Case E. True PSD (—) and estimated PSDs using GCV for replications 2 (—) and 5 (• • •).


100

POLYMER

true PSD

j\ 11

CHARACTERIZATION

EXPF

rep. 4 rep. 5

if ]


ii

// II

/f ii

ii ii it

f(D)

f.

ii

/1 £1

bi

hi

\

/?

*\

\ y *.. / i

50

1300 D[nm]

2550

Figure 9. Case F. True PSD (—) and estimated PSDs using GCV for replications 4 (—) and 5 (• •

Figure 10. Case G. True PSD (—) and estimated PSDs using GCV for replications 1 (—) and 5 (•••).


6.

101


ELIQABE & GARCIA-RUBIO

the o p t i m a l estimates o f the P S D s are not p l o t t e d , they w e r e v e r y close to those o b t a i n e d w i t h t h e 7

values. I n case D ( F i g u r e 7) a m i l d oscillation

G C V

is p r e s e n t i n t h e solution o b t a i n e d for r e p l i c a t i o n 3. T h i s result m a y b e e x p e c t e d w h e n replications are c a r r i e d out. T h e possibilities o f this k i n d o f results are e v e n h i g h e r w h e n , as i n this case, h i g h levels o f noise are present.

Summary and Conclusions Downloaded by UCSF LIB CKM RSCS MGMT on November 19, 2014 | http://pubs.acs.org Publication Date: May 5, 1990 | doi: 10.1021/ba-1990-0227.ch006

T h e results p r e s e n t e d i n this c h a p t e r verify t h e p o t e n t i a l a n d v e r s a t i l i t y o f the r e g u l a r i z a t i o n t e c h n i q u e w h e n it is u s e d along w i t h the G C V t e c h n i q u e . A c o m p l e t e l y different p r o b l e m from those a n a l y z e d i n refs. 18 a n d 2 2 u s i n g the same c o m b i n a t i o n was s o l v e d w i t h few a n d p r e d i c t a b l e l i m i t a t i o n s . T h e use o f these c o m p l i m e n t a r y techniques was d e m o n s t r a t e d t h r o u g h the r e c o v e r y o f the P S D o f p o l y s t y r e n e latices. U n i m o d a l a n d b i m o d a l P S D s of v a r y i n g b r e a d t h a n d m e a n p a r t i c l e diameters w e r e investigated. T h e r e sults w e r e m o s t l y satisfactory. T h e G C V t e c h n i q u e makes it possible to integrate t h e c o m p l e t e e s t i m a t i o n process i n a single step for the p u r p o s e o f m o n i t o r i n g a n d c o n t r o l l i n g a v a r i e t y o f heterogeneous systems, a m o n g t h e m e m u l s i o n p o l y m e r i z a t i o n s .

Appendix F o r t h e d i s c r e t i z a t i o n o f e q 4, i t is assumed that, for a g i v e n w a v e l e n g t h X , the integrand can be approximated b y the product of a linear interpo0 {

lation b e t w e e n t w o successive points o n f(D), fj

f

j+l

w i t h fj = j(Dj), c a l c u l a t e d at \ , i 0

and D

J+1

BJDJ

(Al)

= Aj + BjD

(A2)

=

Aj

+

j+l

- D = A D for a l l j, a n d t h e k e r n e l K ( X , D ) 0

:

K ( A o , , D ) = UD)

(A3)

= \ Q (Xo,i,D)D

2

ext

D i v i d i n g t h e i n t e g r a l i n e q 4 i n t o n - 1 sections a n d s u b s t i t u t i n g i n eqs A 1 - A 3 s h o w that T c a n b e expressed i n the f o l l o w i n g f o r m : {

-

J K (D)(A i

1

Di

+ BJ)) dD + J KJ(D)(A

2

...

2

Di

D

j+

+

+ B D) dD +

g

D

j Dj

N

KJPXAj

+ BjD) dD + ... +

|

Ki(D)(A -i N

+ B ^D) N

dD

DN-I


(A4)

102


w h e r e i t was assumed that f(D) = 0 for D > D > D . T h e values for t h e parameters A a n d Bj c a n b e o b t a i n e d from eqs A l a n d A 2 Y

n

j

D f, - Djf - Dj l+1

1

j

D

(A6)

- Dj

j+1


(A5)

i+1

S u b s t i t u t i n g e q s A 5 a n d A 6 i n t o e q A 4 yields

^

j Kj(D) dD -

±

Di

J

D\

Dz

m

)

d

D

+

ib!

Di

°i

L

K i ( D ) d D

~iD!

^

D

D

)

D

d

D

D

2

2

Dv-i

^

WP)D dD

Ds

D3

~ i b !

+

f i + J f i j

j HD)DdD

DJV-I

UD)DdD-^

J

+ j^

K,{D)dD

J

Dx-2

DN-Z

j

K,{D)dD

DN-I

Dx

~X5 I

^

K

D.v-i

D

d

f-

D

N

Dy

-

\

D

N

U

j

D ) D d D - ^

D.v-i

(A7)

D)dD

U

D.v-i

Therefore a is g i v e n b y tj

1 d i J

=

D

A D /

+

U

AD

D

j"

)

D d

D

"

KID) dD

Di

AD /

^

(

D

)

rfD

- ^ J ^(0)0

(A8)

f o r j = 2, . . . . n - 1, a n d D2

«.I = ^

= ^

/ K / D ) dD -

J D -i N

Ki(D)D d

J K#J)D d D

D

-

^ j

KID) dD

DN-I


(A9)

(A10)

6.

ELieABE & GARCIA-RUBIO


103

F o r a s m a l l i n c r e m e n t i n t h e diameters, t h e integration o f t h e functions K / D ) c a n b e calculated b y u s i n g a straight-line a p p r o x i m a t i o n a n d t h e same step A D u s e d w i t h f(D). T h u s , t h e integrals i n eqs A 8 - A 1 0 can b e w r i t t e n as Dfc+i

K0)

J


Dk

dD « D

4 + 1

2(D

- K*)

t + 1

K , ( D ) D dD » | K * ( D *

J

D

DtKft+i

K» -

+ 1

2

-

(All)

(D*

+ 1

2

-

Dfc ) 2

D* )

(A12)

2

k

2(D*

+ 1

- Djt)

3(D

k+l

-

(D * k+1

-

K ) 2

k

D) k

where K = K (D ). B y s u b s t i t u t i n g eqs A l l a n d A 1 2 into eqs A 8 - A 1 0 , the appropriate values for a c a n b e o b t a i n e d . ik

t

k

tJ

Acknowledgments T h i s research was s u p p o r t e d b y N a t i o n a l Science F o u n d a t i o n G r a n t s R I I 8507956 a n d I N T - 8 6 0 2 5 7 8 . G u i l l e r m o E l i c a b e holds a scholarship f r o m C o n sejo N a c i o n a l d e Investigaciones Cientificas y Tecnicas d e l a R e p u b l i c a A r gentina.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

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104


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for review February 14, 1989.

ACCEPTED

revised manuscript August 1,

1989


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