16: MechanismStudies on the light-focusing plastic rod.of gradient-index formation in photocopolymerization ofmultiple monomer systems

Yasuji Ohtsuka and Yasuhiro Koike

In this paper, we propose a general mechanism for forming a radial GRIN rod by photocopolymerizationin multiple monomer systems and a method of selecting new monomer systems applicable to GRIN. Byusing the simulation with various monomer systems, it is shown that the index distribution of a plastic GRINrod can be tightly controlled by the selection of a ternary monomer system and the photocopolymerizationcondition.

1. Introduction

We have reported that a plastic GRIN rod (light-focusing plastic rod, LFR) could be fabricated by twodifferent processes: two-step copolymerization1 andphotocopolymerization. 2 No other processes have beenreported with the exception of several patents.3 TheLFR prepared by two-step copolymerization cannot beconverted into a filamentary fiber because of its networkstructure. On the other hand, the LFR prepared byphotocopolymerization can be easily heat-drawn intoa plastic GRIN fiber4 in view of its linear structure. Inaddition, the preparation method is simple comparedwith the former.

To control the refractive-index distribution of theGRIN rod prepared by photocopolymerization, theauthors propose a general mechanism for forming theradial GRIN in the photocopolymerization of multiplemonomer systems.

11. Copolymerization of the Multiple MonomerSystem

-M. + Mi X ,M-.-Mi + Mi - Mim-i (1)

(polymer\ (monomer)(radicalJ

(polymer(radical

The authors are with Keio University, Faculty of Science & Tech-nology, Department of Applied Chemistry, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama-shi, 223, Japan.

Received 12 October 1983.0003-6935/84/111774-05$02.00/0© 1984 Optical Society of America.

Equation (1) shows the propagation reactions of co-polymerization between Mi and Mj monomers, wherekij and kii are the propagation rate constants. Here themonomer reactivity ratio rij is defined as

(2)

In the multiple monomer system (M1 M2, .. M,) thedifferential equation of the copolymer compositionis 5

d[Mil= (DE [MI]/ D Ed [A'Ij] k=1 rik h=i rjk

(3)

Here, d[Mi]/d[Mj] is the molar ratio of the Mi monomerunit to the Mj monomer unit in the copolymer formedat any instant, when the concentration of each mono-mer (M1,M2,. . . Mn) in the monomer mixture is[M1J,[M2] ... [Mn] (mole), respectively. The rk (rJk)is the monomer reactivity ratio between Mi (Mj) andMk monomer, and rii = rjj = 1. The small determinantis Dii(Djj) where the i line and i column (j line and jcolumn) have been omitted from determinant D:

n[ , - [M k]/ i I nkc T(I = r1/ /

[M2]rM2

I

[Mn]Irin

(M2]_ M]_ [M 2 ]

k=' 2k r, 2-__I

1Mn ]-Iii2 Mk -_f r_/Mn - Z

1774 APPLIED OPTICS / Vol. 23, No. 11 / 1 June 1984

(4)

(Mi I M ]r2l rn i

i = 1,2 .... n.

rij = kiilkij = 1,2 .... n -

i 51� i

The weight fraction (Yk) of the Mk monomer unit in thecopolymer formed is expressed as

Yk = mmd [Mk]/ Y mkd[MkI,k=1

(5)

where mk is the molecular weight of the Mk monomer.Since the amount of monomer reactivity ratio is n (n -1), a ternary monomer system has many more param-eters than a binary system. As shown below, the radialindex distribution in the former is more feasibly fabri-cated than in the latter.

By solving the differential Eqs. (3) through numericalcalculation and relating it to converted P from themonomer to the polymer, we can analyze the change incopolymer composition during the copolymerizationprocess of a multiple monomer system. The weightfraction (Xk) of the Mk monomer (k = 1,2,... n) in theremaining monomer mixture at converted P is ex-pressed as

XkO- fYkdP(6)Xk= X

where Xko denotes the weight fraction of Mk in the ini-tial monomer mixture. From Eqs. (2)-(6), the co-polymer composition of the instantaneous copolymerformed at converted P is estimated. Furthermore, byusing the Lorentz-Lorenz equation and assuming theadditivity of the molar volumes of structural units, therefractive index n of the copolymer, composed ofd [Mi],d [M2],... d [Mn], is expressed as

/1 + 2ep

\1 -o (7)

n (2 -1 mkd[Mkl / (mkd[Mk]l

k= n + 2 Ppk k=1 Ppk I

where nk and Ppk are the refractive index and densityof the Mk homopolymer, respectively.

The variation of the calculated terpolymer compo-sition with converted P in the methyl methacrylate(MMA)-benzyl acrylate (BzA)-vinyl benzoate (VB)system is shown in Fig. 1, where r1 2 = 2.66, r2 1 = 0.29,r13 = 8.52, r3 1 = 0.07, r 23 = 1.51, and r3 2 = 0.08. Eachmark (0, A, +, and X) expresses the instantaneous co-polymer composition at P = 5k wt. % (k = 1,2, .. ).This trend indicates that MMA richer terpolymer isformed at the initial stage, BzA richer terpolymer at theintermediate stage, and VB richer terpolymer at thefinal stage. The calculated refractive index in thismonomer system is shown in Fig. 2. The refractiveindices of MMA, BzA, and VB homopolymers are 1.49,1.56, and 1.58, respectively. In the binary monomersystem (curve A) the refractive index abruptly increasedaround P = 80 wt. %. Addition of BzA (ternary system)gives a gentle increase in the refractive index as shownin curves B, C, and D.

111. Preparation of a GRIN Rod byPhotocopolymerization

Figure 3 shows the schematic representation of thephotocopolymerization process at a steady state. Amixture of several monomers (M1,M2 ,... M) con-

0 50 100VB ( WTY.

Fig. 1. Change of copolymer composition with converted P in theMMA-BzA-VB system. Each mark (0, A, +, and X) representsinstantaneous copolymer composition atP = 5k wt. % (k = 1,2 .. .

MMA/BzA/VB (wt./wt./wt.): A, 3/0/1; B, 2.5/0.5/1; C, 2/1/1; D,1.5/1.5/1.

1 .60

1.55

C

1 .50

1.450 50 100

P (wt.%)

Fig. 2. Refractive index of MMA- BzA-VB copolymer formed atconverted P. MMA/BzA/VB (wt./wt./wt.): A, 3/0/1; B, 2.5/0.5/1;

C, 2/1/1; D, 1.5/1.5/1.

taining the initiator is placed in a glass tube. Rotatingthe tube on its axis, it is exposed to UV light passingbetween the two shades. The UV source is moved up-ward at a constant velocity. Therefore, the monomeron the dotted line K is just exposed to UV light andphotopolymerization commences. The monomer belowthe line has been exposed for a longer time. When theconversion reaches Pc, a gel phase (with polymer con-tent Pu) is initially formed from the inner wall of theglass tube. The gel phase thickens with irradiationtime. When the content of the glass tube is solidifiedup to the center axis, the conversion reaches Pf.

The copolymer located in the region near the pe-riphery is formed in the early stage of polymerizationand the copolymer near the center axis is formed whenconverted P approaches Pf. Therefore, in the case of

1 June 1984 / Vol. 23, No. 11 / APPLIED OPTICS 1775

0Z

, Glass tube

Upper shading

i thitent Pu 4

4.-UV

4-

4-Fig. 3. Schematic representation of the photoco-

polymerization process in the steady state.

shading

LiquidPhase

be

Gel phase Liquid phase

I1.

PS.4

ci4-,

5 Pu

E

° Pm

Pc

U Sp

Cross-sectional areaof gel phase S

Fig. 4. Formation of the gel phase in a glass tube during the photo-copolymerization process.

curve A (MMA-VB binary monomer system) in Fig. 2,the resulting GRIN rod has a gentle refractive-indexslope in the region near the periphery and a steep dis-tribution in the center region, which are the inherentcharacteristics of the binary monomer system.4 Byincreasing BzA (curves B, C, and D), the index distri-bution of a GRIN rod would gradually approach a par-abolic curve from center to periphery. The details aredescribed in Sec. IV.

IV. Formation of a Radial Gradient IndexAnother representation of the photocopolymerization

process is generated by a computer simulation (see Fig.4). Here, we suppose there is a cylinder of 1-g monomermixture in the glass tube. With the polymer contentat Pc, the infant gel phase with polymer content Puforms on the point of S = 0. When UV irradiation, thegel phase thickens accompanying the polymerization

inside the gel phase. We assume the polymer contentof the liquid phase (Pm) is expressed asPm = PC + (PU -PC) ( the amount of the gel phase ),B

e total amount of the cylinder at P Pf)

(8)

where : is a constant. Further, it is assumed that themonomer mixture with conversion Pm is supplied fromthe upper part of the glass tube into the volume-con-tracted part caused by polymerization.

The dWT and dWp are the weight differentials of thecopolymers formed homogeneously in the liquid phaseand on the inner wall of the gel phase, respectively. Thedifferential of the copolymer further formed inside thegel phase is designated dWg, and dS is the differentialof the cross section of the gel phase formed. Assumingthat the monomer trapped in dS is polymerized in situ,the weight fraction (Xk) of the Mk monomer in the re-maining monomer mixture of liquid phase is expressedby

Xk = Mk/ E Mk,k=l

Mki = Mk(ii)- Yk(js)(AWT + AWp) P - PP - PM(i-l)X Xk(i-,)AWp

+a-a 1 P(i l)]Xk()(AWT + AWp + AWG)1 - o!Pm(i~1 )

(9)

where the subscript i denotes the i step of the numericalcalculation and A expresses the respective infinitesi-mals. The ratio of volume contraction from monomerto polymer, a, is given by

,= n (1l 1 n Y-Yk -_I;/ E -I

k=1 PMk PPk k=l PMk(10)

where PMk and PPk are the densities of the Mk monomerand polymer, respectively. The absolute amount ofpolymer in the liquid phase, Q(g), is given by

Qi = Q(-- AWpPm(i.l) + AWT + aPm(j.l)Pu - Pm(il) 1 - aPm(i-)

X (AWT + AWP + AWG). '(11)

1776 APPLIED OPTICS / Vol. 23, No. 11 / 1 June 1984

Cj

I ----- ---

dWp dWTX __ _ __ _ _ L t>X/~.

I _ _ _ _ _ _ _

82

AWT, AWp, and AWG, which are the functions ofPmykQ, etc., are calculated at each i step of the nu-merical calculation. The yk at each step is calculatedby Eqs. (3)-(5).

After the complete polymerization, the volume (dV)of the shell dS in the cylinder is expressed as

dV dWp (1-P.) x + IP Yk dPmPu-Pm I k=l PPk k=1 Pc PPk

+n X - (1 PC)Xkc] + dW k kEl +~k +dWp> Ik=1 PPk I k=1 PPk

I.53

1 .52(12)

where Xkc denotes the weight fraction of the Ma mono-mer in the remaining monomer mixture of the liquidphase at P = Pc. The resulting copolymer composition(Yk) of the Mk unit formed within the shell dS at 100%conversion is

Yk = (1 - PU)xk + (Xko - C + XkCPc)

pPm+ J ykdPm + (Pu - Pm)yk. (13)

Therefore, the copolymer having the composition of Eq.(13) should be located at a distance r from the centeraxis of the GRIN rod, where r is related to Eq. (12) asfollows:

(r/Rp)2 = 1 - S/Sp,

= 1 - V/Vp,

1.51

(r/R )2

Fig. 5. Comparison of the refractive-index distributions betweensimulations I and II when MMA/AN/VB = 2/1/0.5 (wt./wt./wt.), Pc

= 0.25, Pu = 0.6, Pf = 0.8, and Pm = Pc.

1.0

(14)

where Rp and Sp are the radius and cross section of therod, respectively; Vp is the whole volume of the cylinderdefined above. By using the Lorentz-Lorenz equation,the copolymer composition is converted into the re-fractive index. Finally, the refractive-index distribu-tion of the plastic GRIN rod by the photocopolymeri-zation of the multiple monomer system is obtained.

V. Prediction of the Index Distribution

In Ref. 2 we proposed a mechanism of forming radialGRIN in a ternary monomer system, where the com-plete diffusion of monomer between liquid phase andgel phase was assumed during photocopolymerization.Here, the simulation in Ref. 2 is designated as simula-tion I. And the simulation proposed in Sec. IV is re-ferred to as simulation II. Here, it is assumed that themonomer trapped in the gel phase is polymerized insitu, considering the slow diffusion rate of monomer inthe gel and viscous liquid phases.

Figure 5 shows the comparison of the index distri-butions calculated from simulations I and II when Pc= 0.25, Pu = 0.6, Pf = 0.8, and Pm = Pc. The monomersystem is MMA-acrylonitrile-VB (M1 -M 2-M 3 ) and therefractive index of each homopolymer is n1 = 1.49, n2= 1.52, n3 = 1.58. The monomer reactivity ratios arer12 = 1.34, r21 = 0.12, r13 = 8.52, r31 = 0.07, r 23 = 5.0, andr32 = 0.05. The index distribution of simulation II ismuch steeper than that of simulation I. Here, the as-sumption of Pm = Pc [/ = - in Eq. (8)] is not well fittedto the practical situation. We observed that the liquidphase became more viscous with irradiation time; Pmshould be increased with irradiation time. Figure 6shows the typical variation of Pm. Considering the gel

E0-

Pu

Pc

0 S Sp

Fig. 6. Variation of Pm with the cross-sectional area of gel phase S.

effect in vinyl polymerization and the thermal poly-merization in this process, the case of iB = 2 is moreconceivable than that of j = 1.

Figure 7 shows the effect of:3 in Eq. (8) on the indexdistribution of the MMA-BzA-VB GRIN rod in sim-ulation II. As mentioned in Ref.,6 the computer sim-ulation under 3 = 1.5-2.0 gives a similar index profileto the experimental result. Figure 8 shows the distri-bution of terpolymer composition for the MMA-BzA-VB GRIN rod, calculated by Eqs. (12) and (13), whenPc = 0.25, Pu = 0.6, Pf = 0.8, and : = 2.0. Figure 9shows the effect of monomer feed composition on theindex distribution of the MMA-BzA-VB GRIN rod.The monomer feed compositions are the same as thosein Fig. 2. Here, Pc, Pu, Pf, and / are the same as thoseof Fig. 8. Since the abscissa is (r/Rp)2, the quadratic-index distribution is expressed as a straight line in Fig.9. In the binary monomer system the quadratic dis-tribution was limited to the center region (curve A).With an increase in the ratio of BzA, the index distri-

1 June 1984 / Vol. 23, No. 11 / APPLIED OPTICS 1777

P=oo

.

1 .550

1 .54

C

1 .53

1.52

0 0.5 1.0

(r/R,) 2

Fig. 7. Effect of on the index distribution of the MMA-BzA-VBGRIN rod when MMA/BzA/VB = 2/1/1 (wt./wt./wt.), Pc = 0.25, Pu

= 0.6, and Pf = 0.8. fl: A, 0.5; B, 1.0; C, 2.0; D, 4.0.

100

3

9 50

0 0.5( r / Rp ) 2

1.0

Fig. 8. Calculated distribution of copolymer composition for theMMA-BzA-VB GRIN rod when Pc = 0.25, Pu = 0.6, Pf = 0.8, and

= 2.0. MMA/BzA/VB (wt./wt./wt.): (-), 1.5/1.5/1; (--

2/1/1.

bution was remarkably improved and was in fairagreement with the experimental result.6

After carrying out the above simulation for the vari-ous monomers, MMA-AN-VB and MMA-BzA-VBwere selected as suitable monomer systems in the pho-tocopolymerization. In both monomer systems, theplastic GRIN rod having a quadratic-index distributionup to the periphery was experimentally obtained.6

-0.01

C

C

-0.02

-0.030 0.5 1.0

(r/Rp) 2

Fig. 9. Calculated index distribution for the MMA-BzA-VB rod.MMA/BzA/VB (wt./wt./wt.): A, 3/0/1; B, 2.5/0.5/1; C, 2/1/1; D,

1.5/1.5/1. no is the refractive index at center axis.

VI. Conclusion

To control and predict the index distribution of theplastic GRIN rod, we propose a mechanism for formingradial GRIN in the photocopolymerization of multiplemonomer systems. In the photocopolymerizationprocess, it is assumed that the polymer content Pm inthe liquid phase gradually increases with the thicknessof gel phase, and that the monomer mixture with con-version Pm is supplied from the upper part of the glasstube into the volume-contracted part caused by poly-merization. Here it is also assumed that the monomertrapped in the gel phase is polymerized in situ.

As a result of carrying out the above simulation, it waspredicted that, by extending the monomer system frombinary to ternary, the index distribution of the rod couldbe tightly controlled. According to the calculated re-sults for various ternary monomer systems, MMA-AN-VB and MMA-BzA-VB monomer systems werefinally selected as a suitable monomer system to get aparabolic-index profile of the plastic GRIN rod.

References1. Y. Ohtsuka and T. Sugano, Appl. Opt. 22, 413 (1983).2. Y. Koike and Y. Ohtsuka, Appl. Opt. 22, 418 (1983).3. Japanese Patents (Kokai Tokkyo Koho), 75 83,045; 78 21,937; 82

20,601.4. Y. Koike, Y. Kimoto, and Y. Ohtsuka, Appl. Opt. 21, 1057

(1982).5. C. Walling and E. R. Briggs, J. Am. Chem. Soc. 67, 1774 (1945).6. Y. Koike, H. Hatanaka, and Y. Ohtsuka, Appl. Opt. 23,1779 (1984),

same issue.7. Y. Koike, Y. Kimoto, and Y. Ohtsuka, J. Appl. Polym. Sci. 27, 3253

(1982).

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