te {r. r,....0)

Rz6zów Unive^iiy
of T€ónology,
/'r' Pow*ańĆów wargwy 6, 35 95
.mail hgal@prż rzesżow.pl
A theoretical Fod€l
Deparheni
Industial and Materiak
chmisĘ
of hyperbranched polymeiization involving an
ABlmonomer
Pań
Afrtri"n
oI
r-
DEGREE oF BRAN(]SIN(:
aU !1edi ład lo
fr! t|tol, Pfr|aś Boiną KDlnż on Hd
l
bi\d'
sumnary
A B€neralized theoteti.al model of hyFlbrand]€d polyh.Iiza
lion of an Aq ńolońcr js delvc'l The ńodclis a.laŚi.a]ńeań Iield onc
bascd Ón thc sńoluchowskl .oagulaiion caurhon' ]}e monomer functiolal
8rouPs B react ł(oldi$ b the/ijJl si.ll J'bJł!libń Ź'€d prin.iple' It k shown
lhat iirBPćlively Óf lh. monomer fun.tiÓnality 0' the number av€ra8€
no
leÓlal wei8ht o1hyPdbranched Polyńł9 dcPmd on .onveEiÓn Ó1A grolp!
in eiadly lhe sańe way' Al5o rhe dątć oID''.}łt3, ł paranł.I d*dibin8
Perfc.tneśof Polyńg sfu.forc changft vcry sli€hlly wilh inĆĘasing fun.
Keywolds:n'Tsbrłn.hPdpołTeri2.iioĄkindiĆmodel,sńoluchowski.o'
aSulahon equaiion, substiiuiion dfed, tundionality o( thc mononeE, deSree
An jni.rst in hyperbrancled PÓlymeB has Ę.ently
raPidly gtń'n mo*Iy besus. o1 rh. eiPeded sińilarity
Óf th€ behaviÓlr of hyF€rbnn.hed Folymec ro d.n
d net ańd akodue b Potnha l aPPlica tions ofth. hy.
Pgbla n chcd Po lymcn as Polymcr mod ińcs and mll h.
Iunctioml dcŚlinking a8enh n_zl
In frd ihe ideł of usjn3 nuliifuncti
Polymcrizaiion th Podue hĘhIy bran.h€d' bui nol
ao$łinked matłia] is łnown si
wolks by rbry lĄ 9l who derived an €xpliĆit fundirn
fol ihe size distibltion in FoIyń
nonoh.. Mor. Ęeńtly, the sańe pbbleń Was tackled
by sevdal authoB, indud,ng thc prescnt onc 110-121.
Th6 prp dcaLs wiih ihc kinehc no,lel of polymeriztu
tion of an AUr nonomq with arbibary /. ]n ptuiicular,
thc dcgPe of bi.nching in nodel hyperblan.hed pÓly
b iÓuÓw rhe Ćnan36 in th
.Ul€s Óf diffetent lsĆtivity Gathd than diffdent sŁe or
st!dĘ) it js n..ssa!y usc a P
'o
sin€ the ńonomcl oBidc'cd
rcacb with thc "/'ś5'El]
]n order
słbsiiul''.'.d"(4 d'
n3]),a mPlhod of.odi$ moIe_
Ćul6 ha3 b bc lpPIied, similar io ihat usPd in pr€viou
PaPłs lr5,14l' Trus' each moleu
forń oI ftompon€nr vedori ś= G[ ł,.'., ł J), lh.
enhy s' of whiĆh is dt nunbeG Óf !niL with dadly i B
Eroups alrsdy reachd Notc rhat rhc unib rth alll
8roups B ftaĆEd ale not.ounted and lhat thele B no
oneb.one .ÓrtPondence b€fueeń a smgle vs'or and
the stu.tuft of ib corcspondms mote.ule one vedor
may d*dibe dE sh.brc of many isońcńc Ęoleolcs,
bui tney all sharc ihe sme remiiv y of thcir B grcups
Eiamples ol smll nole.ules thai a
r^
1'
/ '_a
,u ,.śB\<?-U
R^t'
^ś
^
l-te \
{r.
slhdt
^
Efuńpb
r,....0)
o|
hvpłb nctd
'-r' {a
'ir
l,,r.r..ol
obcubs ąnd
rhri
lem and their .orresPonding coń
shÓwn in s.heme A' only agcli. molml.s arc consi_
!LATIONEAUEION
Theonsińlsńoh.nowskiequaiion l1ól is in Iłd an
infinitc sct oI diflcr.ntial eqlatio$ de$dbin8 d'€ rabs
ai which ł nolc.ulc.onsisdn8 of
identical unib (an im.I) n
olt of nvo ńakhing
sn3ll.r ĆÓnPon.nb and lhemle
'ormcdal whi.h thń molc.ule
vanishes
the sysfum in rea.tiolT wilh oihs molc.
'Ón
se.ond
o be ih€
odc'
thc sańc r.ac6!e grÓlPs are the same l13l Analogons
wtbn lo! FolyńdizatiÓn Ó{ an
ABrmonomer From ihesum of Et
tions,atwhichans moleolc is
thc ńlcs of!ea.
'omcd,
tioN are subtacted, at whi.h this
moleculc is clińlnaied
foń the syśeń All drc rats e produch ol apPoPn_
nś. To wrile down all
these rłies is quii! lcdious an opsahon, bul orheńljse
is śtłi8hilotrard'Thcn' ińe Product ol duńńy vari
3bleŚnultiPliest]rer€suxingnte€qu.iion
iŚ
Ónkii€ntł
ńolerles
aPĘśedin
numbel Ólunib in lhe sy cm. Thcftdu€d tińe. is the
a.foal iine muliiPlied by Ił0 The oihel synbols u€d in
e{]U ion (3)
Ą i5 tn. f!ńction H
I' ll=ll K]
K'
whete th.
en.€ pÓiiti lł],
(1)
involving
vańable
vanisns
fńm
dutńy
Produd
łł
Il,,
l(()
lK,,'
I
r)
K=]:],
i=lż
/_]
a)
ibin8 the ratio ol the
rcł.tivit/ of B lroups ń the unir
re.ded (kl) and ih€ rc Ńity of B g'ouPs in
Finallt ihe rab equaiioN ror
pvery
sj
a
tcńinal
e
sumncd
!P. A1łer śÓńE el.menlary al8ebrai. opeia ons one
bin5thefdlo ingsnol!ĆhowŚkilikeequatjÓn]
,H I(x ! AH
TH
)-+r.")-+
-=t
^H
ob_
)HI
*,f.#,-,,#,,,-, ",,#1,
.
//'
ąo)
=
= r and
hÓn degrce
'n
''G)=
iź ' i
n
hal deńvatives of H with
ihc nole.ul€s
ar uscd ro d.notc
sh. ng thc samc code
adopred,
r' ., I r),
(hcy
lun.hons
irc
Point
rs!.d
ł]so .al.ulahd ai ihe r€lerence
lo I1 (i = 0,
alrcaay nention€d' lhe smolu.nowskllikc cquł
(3) desribe rhe rime .vol!rion or the entirc mole.uiion^s
lar size disńbuiion ln ihe polyńelization systeń' Atl
ńoledla!Paft m.t$sbccomcavailable by solvińg oldi
nary d Imcntiłl equa tioB
e d.rivcd Ircm cq. (3)
Hmc wc demoNbat€ lhe n€i]]ods ol Pxbł.hnc ihc .on'
vereion degree, nJmber av.ta8e Polym.rizałion de
gres' and lhe ńodiIicd d*tee o1branĆhing
T'. ÓnvłsjoD dcgrec (P) n defined dif.rently than
in rhe orisinal works of Flory l9l o! zifl n7l t n the
frżĆtion of all A grouPs that havc reacbd Th!s' p varies
60ń 0io 1' noi fiom 0 br/las in thcoń8inald.ńnition
lhłi
The
rnversion degre..an b..a].ulak'l by makiĘ
lÓws lrom el]' (5), HP is ihc
and ońe
rea.td
B grcuP
nUńbd
nducs
of
molelles
Pet Unil,
the nuńbe! of
.uhs by onq rhe .onveEion degree is sinply
Hł k obtained ditdly foń
liAł jntothe.rd niryd fl
bvslb5hfuhnl |Ji
]', l
(3)
e.1.
G) by Ćonvc ing lhe
enńŹl eq
]b\ ]l,
be.ones:
ńole
rr
' rl
ńall
\,
Thc
)
sińila y, thc lundions on lhc €hLhand{ide ofeq
(7) ate av lable
Ę solvm8 lhe syscń (3)of djffdentjal
equa tions obianr€d by diffeienliating €q
wiih rsP( t
rr..
r,rrh'
,r.rlby rr
rLr
rr
by subshruri'ra
'
ThetaderńiEhtvishtoveilvthar:
(4)
and thc bnckc6
lnib
i,= H,(8"+H"+ 4.
wherc ihe .ouniing (genelaiiĘ) fun.tjon H, conhining
, r)=LL )r\,
l,'., '.'."/''r
r/H/ 5 ńe numb*óvera8e polyńłiŻi
thcłśhmTlrefunch'{śH' iI.ńePar
lundions. Tnei\ €q
-,,(,",,.-.,.,)
\
l'l
"",
ca.h iyPe oi unit .an bc
it
ęń
shadnE thc sańc codc dlvided by the tobl
on.entańon
L
i,
EńM' d,fh* M- d -ńJ d_'ń'* ]4
dld
Mń rspedbdtmy tnabLl l followcd by
cvaluatińF t}eir fun.tional lorns in ih€ rcfEenc€
point
yrld noE ordiMry dirier.niial equa
11, *f',..., kr,r
'l
iions wilh.esPed to time for rhe f!ńÓons su.h as, ?'3.
lJ.... ńil are nRded for Gl.ul'ńng w.]8hljvEżge
polińefułmn deg@ U' mommts of
dEb'buhon
and will nor he Ćosidćrcd in tl s papei'zc
LddEnÓleĆll€s=c!,ł,
Eo.a
sr) hm polymmization
$€ code numbeF and
degd.
thc st P śśbi8htfoMłd'As one .an €asily verif,
any hyPclbnnchcd mole.ule of siże ? PreFaied JrÓm
AB/monomo hs r(- r)+r uNcactd B grotPŚ' on th.
oth€r hand, ihg numbelof frcc B 8rouF is śdl+ nf ])+
'
(r')' The Ęlatioń b.rween
'
P-tr- ) G,,J+\tJ
')+
M,=:'
+"/
)
r
oo)
': tł,.s.....J, ]P'
where le is Biven by Rl. (10) and the square blackeb
denole lhe .on.enbalion ol thc Eolcolcs sha!ńg the
Nde thł dre
2i}$
mom€rt
is
bhl nuńbł Óf uńb
j6t H/
l/Pl
ib in lhe nol€
' Thc
ftc
sbi*in€
k|' the d€iinilion
l€s
of
Ón..nbation !nib), hen.e, MI = 1
mcnt M: śthe wei8ht average Polyńeńzation deEiee in
the sy*cm, l,p' Ił is noi diffiolt b sec that .ońPońcnb
of ihc sun in .9. 01) for t = r and 2 havc rhc lorm sJ lso
. ,,tsLl an,l, ,r1.0 . .,.tsr1 r6pd'eLy /, i= I 2,
' '' l rl' These pmdulE ire e\prś'Pdbv K H Jnd
divided by rhe
.ondusion is that thc numbm avqage
t givcn by lhc familńr cqua
de8rcc of Pol)mc zahon
"l
in sprc of ihe numbn of B groups in ABt monoms (t.
prcvidcd ihe .onveEion n cxpr6scd as thc fnction ol A
gEuF thal hav.read€d k|' e9 (ó)l.
1lre lim. oi .onv€rsion evolulion o' th. w.i8hi aveF
ag. Polyńer'atioń d*re. do6 dep.nd Ón
but tnis
analy9is will b€ dealt włh in a fońh.ońing paper
'
(D') in hyPerbranĆhed
'The dcgrcc of blancning
was
k/ €3 rel n3l)a\;
mcr
mol@
lcs
intodu.ed
Po\
ńcasurc oI ihff brJnchjns p.rfu
rmnon ABrrype polyme$ it was dcfined as turc the
Fi3 1 DeyĘ ł hafthin& (DB) !Ś landŚb ł a| A
pa|ps in pal1htlizłlbl o| ABJ nd|nłs |d | = 2 h),1 lb)
lnd 6 łr) hl rLL s* lhc rulc anlrnl o| rYdi,n b.funr
ślÓlp dtd lh| |nsl B ira|p i r |nł, tł = 1' Ft E e|nć
^ b bd,ń lherute nlrll a|,ht lndinl o| u ndl
tÓn kp
B 2lduw uls ekln b b2 k|= 3kr' h= 2k| tk|=k,1t ńar)
Ry), Ę ź k! 2, rhd |i=
kil3
number of bran hin8 (denddti.) unic, divided by ile
total nunbł o1lnits in the syst€n' The d$re. of
bran.hing gnouB be żer for l0eai polymeE and eq!ó]
to one 1or a Pc cd dcndlinc. ThĆ DB is rhe quanhly
that .an bc datively e$ily łalualcd by NMR sPmbo'
In rhe randon polymdiation or an AB, monomer
(j'.', il the sy*em wh€r. nÓnÓnł
siitution effrt) d'. branding ind.x
ÓnvEsiÓn Óf A gtfuPs and Ęa.hes /, a l p aP Ploach_
L
l monomer remiing with PciHr sub.
stibliÓn .fiat, i?', dP nonomer where lhe second B
gi@P ĘaĆ6 faster inan in. fiti I10]'
Fo! I hi8her t\an 2 tne definition or rn. DB was pr&
vided by Hólh! d al' t]3l. In ldńśof the synbol9 E.d
in ink Papel, ihe d€8lge of branchin8 is sińPly ex_
ĆreasPd by usin3
n H
Lś I
''H]"
Tne rśulb o' caldlation of ihg DB
monomen
ABł ABr, and A86 ar€ Prśenledjn Fj8 'oI
1 The .Wgs
representing ine dange or ine de8ree or b.ancning with
convelsion of A glolPs were obtainad aŚuning all
B glouF to leact at the sańe iate (niddle orv. in .aĆh
Plot, Ęndot !ca.hon) The .ury6 j!* above ihe ńid
dlc oną and the hiEhst, wge obtained by bking every
next B 8IouP io r€act Mi.e oI ihrec tińes fa* tha! the
-previous" gmup, resppdively (psitive sub*itution
.1f€Ć')' The cutre just b.low &. tiddl€ one and lhe
lowe* were obhined hkin8 rhe n€8aiiv€ substibtion
effect, rhele each ne* B soup in d'
thrce tines slowcr than rhe prcviols one,Bpetively.
As onc .an seą in all Plob, the deglee o1branĆnińE
lineally tlepen.ls on rnveiśon łnd lhe liniiing dggle
oIblan.hin€ at fu]l rnvcśio n sli8h lly dc cascs wilh
Iumtion.Iily f Tnis bsl con.lusion a8r€ with publish.
wi& rhe Prsent
calolaE thc degEe
of bńrching with
]t se.ns worL\ pÓiniing Óut d'ai
ńodel ł j5 not ńc.e$a!y anyńÓr. lo
of blanchlng to a$Ś Pe ehs3
.on.enbahons o' unib oI diffłent subśifulion degreg
ihłt ar avłilable foom the pr€ent model at each sag€ of
rea.tion, ore Gn emily
wdt
the
lyńeńżhon prclB ńvolving
1'
3.
4.
oure
AB' nonomcr
MalmstÓb E., Hulr A'|
16l
pd
r99ó'
2'
or any
M.ffioJelk
firlkllr:1994,27'1611'
yan D.,
B€dnatk
zlfr!
z.:
t'ałdwlr!]cs
1999, 3r- a19.
M ] Polł'ły 200' a& 163
6 sękD.: Palńlr! 200L 41,757
7 calim H., Lehowie J. B. ?'l;'?/y 200r,4ó' 340
3' r'rl p']'| I. Ań' chzn sk'1%2,7a,zz1s'
9' Fuy ł' ]'| ?ra.ł:t* o| potyht clTnislt!", cJaFl 9
"Md{nkl urrhl dn!ńn!ful i n klłąŃr PlbW\
!frd
lhŻ lńr!
d|
łrlaldf, ctttl]
uniu.
PlB, IlhM
śmńxky J', snń*oń M', simotri.k w.
, ,wleekL': Palyml B|lłlir 1999,4z.Ą9
!' caliM H., LRhowja l' B'' Kaenmki ( futrłdl
ro. Duśek K,
|
r2 cłIjna H', kĆhÓ i.Ż I
(h.Ę://rłreePolymeF oĘ)'
B cahńH,Lchowie' B:l'' Prlln sti1$,1l7'
15' calina H.,
L..howię J
B.,
l,$
1ó' sndlcnowski M'v]
z. 1916, 1z 557'
17' zilIR. M.| 1' sbl' Ph!ś1930,2l'241'
13 Hóllł D ' BuĘath A', rl€y H
:,.4