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Homodyne dynamic light scattering in supramolecular
polymer solutions: anomalous oscillations in intensity
correlation function
Emilie Moulin, Irina Nyrkova, Nicolas Giuseppone, Alexander Semenov, Eric
Buhler
To cite this version:
Emilie Moulin, Irina Nyrkova, Nicolas Giuseppone, Alexander Semenov, Eric Buhler. Homodyne
dynamic light scattering in supramolecular polymer solutions: anomalous oscillations in
inten-sity correlation function. Soft Matter, Royal Society of Chemistry, 2020, 16 (12), pp.2971-2993.
�10.1039/c9sm02480h�. �hal-02905292�
Supramole ular Polymer Solutions: Anomalous
Os illations in Intensity Correlation Fun tion
Emilie Moulin, 1;y Irina A.Nyrkova, 1;y Ni olas Giuseppone, 1 Alexander N.Semenov, 1; Eri Buhler 2; 1
Institut CharlesSadron(ICS), CNRS -UPR 22,Universitede Strasbourg,
23 rue du Loess, BP84047, 67034 StrasbourgCedex 2, Fran e
2
Matiereet SystemesComplexes(MSC), UMR7057, Universitede Paris (UniversiteParis
Diderot), B^atiment Condor et,10 rue Ali eDomon et LeonieDuquet, 75205 Paris Cedex13,
Fran e
(January31, 2020)
y
Contributed equally to this work
Corresp onding authors:
alexander.semenovi s- nrs.unistra.fr, eri .buhleruniv-paris-diderot.fr
Abstra t
Dilutesolutionsofele troni allya tivemole ules apableof
irradiation-driven supramole ular self-assembly are studied by dynami light
s atter-ing. We dete tunusual well-de ned os illations in the long time range of
thehomo dyneintensity orrelationfun tionforallsolutionsthatwere
irra-diatedwithwhitelightpriortothemeasurements.The os illatione e tis
attributed tothelo al laser-indu ed heatingof thesamples due to
strong-ly enhan ed absorption manifested by the supramole ular laments. It is
found that theos illationfrequen y dep ends on theirradiationtime,
solu-tion on entration, and thein ident laserp ower, butis indep endent ofthe
s atteringangle. Theseobservationsareexplainedwithasemi-quantitative
theoryrelatingtheos illatione e ttothermo-gravitational onve tion ows
generated bylaserb eam. The resultssuggestthatthepresen eof su h
ho-mo dyne os illations ould b e a sensitive prob e for aggregation in many
omplexsystems.
Typ eset using REVT E
Light s attering has provided an imp ortant to ol to hara terize the stru ture,
lo al dynami s and motions of parti les in omplex systems [1℄. Indeed, p olymer
solutions and gels, supramole ular self-assemblies and olloidal disp ersions have b een
routinely studied with laser light s attering for many de ades [1{ 5℄.
Dynami light s attering (DLS) in parti ular turned out to b e a p owerful
te hnique allowing to dis riminate b etween solute parti les of di erent sizes and
to determine the parti le size distribution in a solution or susp ension [1,3,4,6{ 9℄.
This metho d p ermits investigation of mixtures of widely disparate sp e ies whose
relaxation times may di er by many orders of magnitude.
A topi al example for DLS appli ation on erns self-assembly pro esses in
solu-tions of mole ular building blo ks onne ted by non ovalent reversible asso iations
(leading to formation of asso iative p olymers, gels, brils, mi elles, vesi les, and
other supramole ular materials) [1,2℄. Su h systems typi ally involve mesos opi or
ma ros opi stru tures whose omp osition an vary in resp onse to external
stim-uli (temp erature, pH, on entration, light). [10{13℄ The DLS signal is determined
through dynami al orrelations of lo al ele tri p olarizations in the system, and
their relaxation re e ts the kineti s of the omp osition hanges. In parti ular, DLS
proved to b e a very eÆ ient to ol to study self-assembly pro esses initiated by
external triggers [6{9,12{ 14℄.
However, the DLS studies of the evolution in the self-assembling systems are
not always straightforward, as su h systems are typi ally hara terized by
multi-ple length-s ales and time-dep endent nature of resp onsive p olydisp erse assemblies.
Moreover, the asso iating units themselves an ause light absorption, uores en e,
plasmoni and other e e ts ompromising the standard DLS proto ol. To over ome
these problems in studying novel highly omplex self-assembling systems, it is
essen-tial to develop new exp erimental strategies hara terizing the dynami s (in luding
lo al velo ities and mobilities) of all individual sp e ies involved in the system.
A key diÆ ulty met in many supramole ular and ma romole ular systems is
related to the l ight absor ption ausing su h unwanted side e e ts as lo al heating,
onve tion and thermal di usion. This often imp oses restri tion on the on entration
range a essible to the exp eriments. The ompli ations arising from light absorption
have deterred resear hers from studying many omplex systems using light s attering.
The main onsequen es of lo al heating due to absorption are the app earan e of
the so- alled thermal lens (whi h auses the in ident laser b eam to diverge) and the
anomalous os illations arising in the intensity-intensity orrelation fun tion measured
by the homo dyne DLS [15{ 17℄.
In this pap er, we study su h unusual long-lasting r egul ar os il l ations in
ho-mo dyne DLS sp e tra observed in s attering of multi omp onent supramole ular
disp ersions based on self-assembling triarylamines (TAA) forming highly ondu tive
nano-wires up on irradiation [12,13,18{ 22℄. We dis uss the p ossible origins of these
os illations and resp e tive sp e i details of their manifestation. We show that,
despite ompli ations due to light absorption, the os illations in the low-frequen y
sp e trum of the homo dyne DLS an b e used to hara terize the system, in
par-ti ular, to provide information on the drift velo ities of the s attering parti les in
solution. Note that it is the hetero dyne light s attering (laser doppler velo imetry
observation of os illations with a homodyne DLS an provide a simple alternative
to the more involved and ompli ated hetero dyne DLS studies [1℄.
The pap er is organized as follows. In the next se tion we des rib e the
light-absorbing TAA systems and the DLS te hnique employed in the pap er. Se tion
3 presents detailed DLS data on intensity orrelations in TAA solutions involving
unusual os illations in their homo dyne sp e trum. We show that the os illation
frequen y dep ends on the irradiation time, on entration, and the in ident laser
p ower, but is indep endent of the s attering angle. These observations are explained
from a theoreti al standp oint by relating the os illation e e t to thermo-gravitational
onve tion ows generated by the laser b eam. Se tion 4 is devoted to the related
dis ussion and theoreti al analysis of the obtained homo dyne DLS results.
2. Materials and methods
2.1. Formation and properties of TAA nanowires
We ondu t our detailed analysis on solutions of triarylamine (TAA) derivatives
with tailored side-groups that are apable to self-assemble into highly ondu tive
supramole ular nanowires up on white light exp osure [12{ 14,30℄. The nature of
multi-stage o op erative self-organization of TAA entities was elu idated in detail by
a ombination of exp erimental and theoreti al to ols in our previous works [14,30℄.
Some key features are illustrated in Fig. 1 for a typi al example of monoamine
triaryl mole ule (triarylamine). In the present exp eriments we exploited the same
TAA mole ules whose systems have b een well- hara terized for all stages of their
self-assembly pro ess in hloroform [14℄. In parti ular, we showed that the
light-indu ed aggregation is initiated by radi al ations TAA :
+ whi h are formed by
photon-ex itation of the TAA unimers in an oxidating solvent (stage (i) in Fig. 1a).
The radi als (adopting the form of TAA :
+Cl- dip oles) rst aggregate with themselves
(via dip ole-dip ole attra tion) and then with neutral TAA forming the
double- olumnal riti al nu leus by harge transfer in onjun tion with hydrogen b onding
(of amide groups) and sta king of aryl rings (stage (ii)). The short
bril fragments then grow in length via end atta hment of free TAA mole ules
(stage (iii)). Finally, the brils aggregate b oth end-to-end and laterally forming
mi rometer-long and few-nanometer thi k stable bundles serving as nanowires (stage
(iv)). These nanowires, if formed under irradiation, were proved to show high
metalli ele tri ondu tivity; they an indu e plasmoni e e ts and give the
solution a sp e i bright dark green olor (whereas the original unimeri TAA
solution has a light yellow olor).
It is imp ortant to stress that the original unimeri TAA hloroform solutions are
stable in the dark for months. Apart from white-light initiation, the self-assembly
pro ess an b e provoked also by adding a small amount of an oxidant, or by
adding of the so- alled `seeds' (small ultra-soni ated fragments of brillated TAA).
For short light-pulses the self-assembly ould take a signi ant time (up to an hour
for 1s-pulses), but it takes just a few se onds for the light exp osure of 5s. Our
analysis [14℄ showed that initial purely unimeri solutions an self-assemble only
in the presen e of radi als, and that the riti al nu leus for the TAA assembly
the b eginning of the bril formation, well b efore the reservoir of TAA is depleted,
so that the thinnest brils (with no lateral aggregation) may b e exp e ted have
roughly the same length at the end of the self-assembly pro ess [14℄. However, the
solute is typi ally dominated with `wires' of di erent thi kness (double brils and
their lateral sta ks, bundles) whi h, onsidered together, show a signi ant length
p olydisp ersity b e ause the e e tive growth rate dep ends on the bundle thi kness
(and due to rare stru tural defe ts preventing further elongation of an aggregate).
On es formed, the TAA brils are extremely stable, their lo al stru ture was
proved to b e un hanged for many months of in ubation in the dark (a slow growth
of bril length was nevertheless observed due to on atenation of brils). Besides,
the fra tion of the radi als in brils f (originally in the range of f ' 10 20%
after a 1-hour irradiation) slowly de reases during the dark in ubation (within
a time-s ale of 1-2 hours f approa hes 5% and then de ays as f / 1=t),
but it an b e re overed after an extra stage of white-light illumination. The
qualitative b ehavior of the radi al amount is re e ted in the UV-vis absorption
sp e tra, Fig. 1d. While the freshly-formed TAA brils with f > 10% are highly
ondu tive and brightly green up on the irradiation, the same brils with f < 3%
are brownish and less ondu tive. The brils whi h were initiated by seeding or a
short white-light pulse are almost olorless or slightly yellow, similar to the original
unimeri TAA solution.
We established that b oth the radi al loss and re overy take pla e at the bril
ends where the radi als an either b e formed by photon absorption or de ay via
pairwise radi al annihilation. The pairwise nature of the radi al de ay (whi h
demands the simultaneous presen e of two radi als at a bril end) together with
the slow hara ter of radi al di usion (from the middle of a bril to its ends)
ensures the apparent p ersisten e of the radi als (their de ay as f / 1=t is mu h
slower than the ordinary exp onential law).
2.2. Experimental spe i ations
The details of TAA synthesis, puri ation, hloroform solution preparation and
irradiation are des rib ed in refs. [12{ 14℄. White visible light irradiation was done
with a 20W glow- lament lamp lo ated at 5 m from the sample (a glass test
tub e with TAA solution). Absorption sp e tra (Fig. 1d) were re orded using a
UV-Vis-NIR sp e trophotometer Cary 500 s an Varian in quartz glass uvette with
1.0 m opti al path.
The homo dyne dynami light s attering (DLS) measurements were p erformed in
UMR-7057 lab oratory using a 3D DLS sp e trometer (LS Instruments, Frib ourg,
Switzerland) equipp ed with a 25mW HeNe laser (JDS uniphase) op erating at
0
=632.8nm (b eam diameter 0.7mm at 1=e 2
of maximum intensity, b eam divergen e
1.15 mrad), a two- hannel multiple tau- orrelator (1088 hannels in auto orrelation),
a variable-angle dete tion system, and a temp erature- ontrolled index mat hing vat
(LS Instruments); the temp erature was xed to T =293K. The s attering sp e trum
was measured using two single mo de b er dete tors and two high sensitivity APD
dete tors (Perkin Elmer, mo del SPCM-AQR-13-FC). The p ower of the in ident and
transmitted laser b eams was measured using a Thorlabs PM 100 high sensitivity
laser b eam (of diameter d beam
2mm at L1) is fo used on the sample with the
lens L1 (fo al length F 1
=250mm), so that the illuminated volume has ross-se tion
diameter of d s
100m [5℄. The b eam one divergen e angle in the sample is
s d beam =F 1 810 3 (1)
The s attering light is olle ted with the lenses L2 (F 2
= 250mm) and L3 (F
3 =
3:9mm) onto an opti al b er (with the opti al ore diameter d 4
=4m) onne ted
to an eÆ ient avalan he photo dio de dete tor (APD). The lens L2 is pla ed at
distan e x 2
= F
2
from the sample. The s attering volume a essible for the
dete tor is nearly ylindri al with the diameter d s
and the length L s
along the
primary laser b eam,
L s F 2 d 4 =F 3 260m (2)
For the DLS studies, dilute solutions of monomeri TAA were prepared in
deuterated hloroform CDCl 3
and transferred into 5 mm diameter ylindri al
s at-tering ells. Dust and impurities were removed from the samples b efore irradiation
by ltration through 0.22m PTFE Millip ore lters. Prior to irradiation with
visible light, no eviden e for the presen e of self-assemblies or larger obje ts ould
b e dedu ed from the DLS: no hara teristi de ays were observed in the intensity
orrelation fun tion apart from the unimer mo de with relaxation time uni
'3s
orresp onding to the average geometri al size of 1nm ( f. Figs. 1b, ). Moreover,
no aggregates were dete ted after a long (up to 8 hours) DLS run at a maximum
laser intensity (P 0
= 22mW), proving that irradiation at wave-length 0
=632.8nm
do es not indu e any TAA self-assembly.
Figure 1d shows the UV-Vis sp e tra of a 1 mM solution of TAA in hloroform
in the absen e of light irradiation (0s), for samples irradiated for some time (765s,
3765s), and for a solution rst irradiated during t=3765s and then stored in the
dark for 14400s (4h). We observe that the irradiated supramole ular solutions do
absorb light; in parti ular, the absorban e around 633 nm (the laser wavelength
used in our light s attering exp eriments) strongly in reases with the irradiation
time, saturating for t > 3000s. By omparison, in ubation of irradiated solutions
in the dark leads to a de rease of the absorban e. Thus, light absorban e of TAA
solutions strongly dep ends on their aggregation state and irradiation pre-history.
2.3. Homodyne and Heterodyne Dynami Light S attering (DLS)
te hniques
The setup of Fig. 2 implies irradiation of the sample with a V-p olarized laser
light. The s attering ve tor is
qk 2 k 1 ; q= 4n 0 sin 2 ! (3) where k 1 and k 2
are wave-ve tors of the in ident and s attered light (whi h are
b oth parallel to a horizontal plane), n is refra tive index of the sample (n=1:445
at T =20 0
C for pure CDCl 3
thermal sp eeds (v T
) is typi ally quasi-elasti (as v T . 100m/s l = 310 8 m/s,
the velo ity of light), hen e k 2 =k 1 =k =2n= 0 .
The s attered ele tri eld E tr ue
(t)=<[Eexp(i!t)℄ 1
at the dete tor is determined
by the a tual p ositions of the s attering enters r m
(t) present in the s attering
volume V: E / V (q ;t) 0 X m exp iqr m (t) (4) where V
(q ;t) is the Fourier transform of the parti le on entration V
(r ;t)
al ulat-ed over the s attering volume V (here and b elow the prime over summation means
that it in ludes only those parti les that are present in the s attering volume V
at the given time t). Eq. (4) is valid in the quasi-elasti approximation, that is, if
the s attering enters move slowly with sp eed v T
l
=L s
, whi h is typi ally true
for soft matter studies (here L s
0:1 1mm is the linear size of the s attering
volume).
The instantaneous intensity of the s attered light is I tr ue (t)= l E tr ue (t) 2 =4, while
the mean intensity (averaged over the os illation p erio d 2=!) is:
I(q;t)= l 8 jEj 2 = l 8 EE / 0 X m 0 X n exp iq[r m (t) r n (t)℄ (5)
When the mutual p ositions of the s attering parti les r m
(t) hange, the eld E,
eq. (4), and the intensity I, eq. (5), hange a ordingly. Hen e, time orrelations
in the slow- hanging ( ompared to the light frequen y) E(t) and I(t) fun tions
arry information ab out the parti le motions.
The following two time- orrelation fun tions are useful for interpretation of DLS
exp eriments: the auto orrelation fun tion of the eld:
G (1) (q; ) l 8 hE (q;0)E(q; )i (6)
and of the intensity:
G (2)
(q; )hI(q;0)I(q; )i (7)
The statisti al averages in eqs. (6), (7) are equivalent to time-averages over a long
enough time, t D LS : h X(0)X( )i= 1 t D LS Z t D LS 0 X(t)X(t+ )dt (8) The fun tion G (2)
, is often alled the homodyne orrelation fun tion, is dire tly
measured by the 2- hannel tau- orrelator onne ted to the photomultiplier light
dete tors of the DLS setup. Eqs. (4), (5), (7) onne t G (2)
with orrelations of
the s attering parti les:
G (2) (q; )/F 2 (q ; ) D V (q ; ) V (q; ) V (q ;0) V (q ;0) E (9) 1
In a ordan e with eq. (4) the fun tion G ( alled the heter odyne orrelation
fun tion) is prop ortional to the dynami stru ture fa tor (auto orrelation fun tion
of on entration u tuations): G (1) (q; )/F 1 (q ; ) D V (q ; ) V (q;0) E (10)
It an b e dire tly measured with the so- alled heter odyne DLS, when a p ortion
the in ident laser b eam (E 0
) is added to the s attered light E, eq. (4), hen e
the resulting eld at the dete tor will b e E tr ue
(t)=<[(E 0
+E)exp(i!t)℄, where E 0
in ludes the phase shift. The DLS-measured ount-rate auto orrelation fun tion is
then G (2) het (q; )hI het (q; )I het (q;0)i (11) where I het (t) =( l =8)jE 0 +E(t)j 2
, f. eq. (5). With no u tuations of the added
light (E 0 (t)=E 0 = onst), we get G (2) het (q; )'I 2 0 +2I 0 G (1) (q;0)+2I 0 < h G (1) (q; ) i +G (2) (q; ) (12) where I 0 = l jE 0 j 2
=8 is the added light intensity and the terms involving E(0)E( ),
or those linear or ubi in E are omitted as they vanish for d s
1=q. If the
added in ident eld E 0
dominates the s attered signal: jE 0
j jEj, the last
quadrati term in eq. (12) an b e negle ted, hen e
G (2) het (q; )' A+< h G (1) (q; ) i =B
where A and B are onstants (indep endent of ). Therefore indeed the hetero dyne
DLS allows for determination of the real part of the hetero dyne orrelation fun tion:
< h G (1) (q; ) i =B G (2) het (q; ) G (2) het (q;1) (13) sin e G (1)
(q;1)=0 for a system in the liquid state.
In pra ti e the DLS exp eriments are often des rib ed using the normalized
hetero dyne and homo dyne orrelation fun tions ( f. eqs. (6), (7)):
g (1) (q; ) G (1) (q; ) G (1) (q;0) = F 1 (q; ) F 1 (q;0) (14) < h g (1) (q; ) i = G (2) het (q; ) G (2) het (q;1) G (2) het (q;0) G (2) het (q;1) (15) g (2) (q; )
h I(q; )I(q;0)i
hI(q;0)i 2 = G (2) (q; ) G (2) (q;1) = F 2 (q; ) F 2 (q;1) (16) Hen e, g (1) (q;0) = 1 and g (2)
(q;1) = 1 by de nition. Note that G (1) (q;0) = l 8 D jE(q)j 2 E
I(q) gives the angular dep enden e of the time-averaged s attering
intensity (here I(q) = h I(q;t)i, where I(q;t) is de ned in eq. (5)). Similarly,
G (2) (q;1)=I(q) 2 , G (2) het (q;1)=I het (q) 2 (I het
(q) is de ned in analogy with I(q)), so
g (2)
(q; )
hI(q; )I(q;0)i
I(q) 2
ases
For illustration purp oses we present b elow the relationships b etween the DLS
orrelation fun tions and the self-intermediate s attering fun tion for some simple
mo del systems.
Let us de ne the parti le status fun tion [1℄
b m (t) ( 1; if m2V 0; if m2=V (18)
whi h shows if a parti le m b elongs to the s attering volume V at time t: Then
V (q ;t)= X m b m (t)exp iqr m (t) ; (19) V (0;t)= X m b m (t)=N(t) (20)
where N(t) is the urrent numb er of parti les in V: The summations here are now
p erformed over the whole system ( f. (4)). Next, we de ne the self-intermediate
s attering fun tion F s (q; ) D exp iq[r m ( +t) r m (t)℄ E (21)
whi h hara terizes the displa ement of a parti le during time . It is imp ortant
to stress that we onsider the stationar y systems only in this study: any joint
probability distribution do es not hange with a time shift t, so F s dep ends on , but not on t. For q6=0 F s (q ;0)=1 and F s (q ;1)=0 (22)
2.4.a. Ideal ase: dilute solution of identi alpoint-like parti les
If the s attering parti les do not intera t dire tly with ea h other (the solution
is dilute enough), their traje tories are statisti ally indep endent. Hen e,
F 1 (q; )= * X m b m ( )b m (0)exp iq[r m ( ) r m (0)℄ + (23)
f. (10), (19) and note that in the light-s attering exp eriments the zero s attering
wave-ve tor is always ex luded: q 6=0, hen e D
exp(iqr ) E
=0. We also note that
the exp onential fa tor in eq. (23) re e ts the statisti al prop erties of the parti le
displa ement r m ( ) r m ( ) r m
(0) whi h are exp e ted to b e the same for all
parti les in the s attering volume V: It is obvious that b m ( )b m (0) and r m ( )
are statisti ally nearly indep endent if
jr m ( )j L s (24) so
F 1 (q ; )' X m hb m ( )b m (0)i exp iq[r m ( ) r m (0)℄ (25)
There is no need to satisfy the ondition (24) for large jr m
( )j 1=q sin e
in this ase the exp onential fa tor gets very small, hen e this ondition an b e
repla ed by
1=q L s
The latter ondition is typi ally satis ed in the DLS exp eriments. 2
Thus, eq. (23)
for the hetero dyne orrelation fun tion (6) transforms into
F 1
(q ; )'h NiF s
(q ; ) (26)
where we used eqs. (20), (21) and h Ni hN(t)i is the statisti al average of the
numb er of parti les in V [1℄ (note also that b m
( )=b m
(0) for most of the parti les
sin e jr m
( )j L s
).
From the statisti al indep enden e b etween the s attering parti les and b
e-tween their r m
( ) displa ements and b m
(t) values we get for the fa tor F 2
( f.
eqs. (9), (19)) similarly to eq. (25):
F 2 (q; )= * X m b 2 m ( ) X n b 2 n (0) + + + * X m b m ( )b m (0) X n6=m b n ( )b n (0) + F s (q; )F s ( q ; ) (27)
Then, using eqs. (18), (20), we obtain:
F 2 (q; )=hN( )N(0)i +hN(N 1)i F s (q ; ) 2 (28)
For large systems with N 1 parti les we an negle t u tuations of N, hen e [1℄
F 2 (q; )=hNi 2 1+ F s (q ; ) 2 (29)
(the so- alled Gaussian appr oximation). In this ase the two DLS orrelation
fun tions are onne ted ( f.eqs. (26), (29), (22)):
F 2 (q ; ) F 2 (q;1)= F 1 (q; ) 2 (30)
The normalized orrelation fun tions, eqs. (14), (16), are onne ted in a similar
way: g (2) (q; ) 1= g (1) (q; ) 2 ; = F 1 (q;0) 2 F 2 (q;1) (31)
(the so- alled Sieger t r el ation) [1℄. From the de nitions eqs. (9), (10):
F 2 (q;1)= h V (q;0); V (q;0)i 2 = F 1 (q;0) 2 ;
hen e =1 for the ideal ase onsidered in the present subse tion.
2
Infa t,thetypi alDLSrangeofwave-ve torsis: 0:510 5 m 1 <q<210 5 m 1 (s attering angle = 30 0 150 0
) and the s attering volume linear sizes are L s
' 0:1 1 mm (Fig. 2),
hen eL s
Let us onsider a dilute solution of identi al di usive parti les whi h are for ed
to move (by an external for e and/or a ma ros opi ow) with an average velo ity
v . Then the parti le on entration (r ;t) follows the di usion equation with a
drift [1℄:
t
+rJ =0; J =v (r ;t) D r (r ;t) (32)
where J(r ;t) is the parti le ux, and D is the di usion o eÆ ient whi h is related
to the so- alled hydro dynami radius R H
of the parti les via the Stokes-Einstein
relation: D = k B T 6R H (33)
where T is the absolute temp erature and is the solvent vis osity. Solving
eqs. (32) we get: (r;t)= Z F s (r r 0 ;t) (r 0 ;0)d 3 r 0 ; (34) F s (r ;t) 1 (4D t) d=2 exp (r v t) 2 4D t ! (35)
where d=3 is the spa e dimension, and (r 0
;0) is the initial distribution.
The Green fun tion F s
(r;t) in eq. (34) (also alled the Van-Hove spa e-time
orrelation fun tion) is a onditional probability to nd the parti le at the p oint
r at time t if initially it was at r=0 (i.e., for the initial ondition (r ;0)Æ(r )):
Hen e F s
(r ;t) simultaneously determines the self-intermediate fun tion F s (q ; ) ( f. eq. (21)) [1℄: F s (q; )= Z e iqr F s (r; )d 3 r (36)
From eqs. (35), (36) we get for a dilute solution with a drift:
F s
(q; )=exp(iqv ) exp( q 2
D ) (37)
From eqs. (14), (26), (37) we get the fun tion < h g (1) (q; ) i whi h an b e
determined in a hetero dyne exp eriment ( f. eq. (15)):
< h g (1) (q ; ) i =< h F s (q ; ) i = os(qv ) exp( q 2 D ) (38)
This is an os illating fun tion with the p erio d
T os = 2 qv (39)
arrying an information ab out the value and the dire tion of the drift velo ity v .
The ab ove equations serve as a basis of the laser Doppler velo imetry [24{ 29℄.
In ontrast to g (1)
, the homo dyne fun tion g (2)
normally has a simple exp onential
shap e and do es not dep end on the drift v: indeed, for many parti les in the
s attering volume h Ni 1 eqs. (16), (29), (37) give
g (2) (q ; ) 1= F s (q ; ) 2 =exp( 2q 2 D ) (40)
Let us turn to the more general ase with several distin t typ es of s attering
parti les in the solution ontains. The amplitude E of the s attered ele tri eld
now is (instead of eq. (4)):
E / X p M p (p) V (q ;t)C V (q;t) (41)
where the summation is taken for all typ es (p) of the parti les, M p
is the material
onstant hara terizing p-parti les, and (p)
V
is the Fourier transform of the spa e
distribution of these parti les (as de ned in eq. (19) with the summation over
p-parti les only). In analogy with se tion 2.3, one an show that the homo dyne
and the hetero dyne DLS exp eriments yield the fun tions < h g (1) i and g (2) de ned
with eqs. (14), (15), (16) in terms of the resp e tive orrelation fun tions F 1 and F 2 ( f. eqs. (10), (9) with C V (q;t) instead of V
(q;t)). We de ne also the partial
orrelation fun tions: F (p) 1 (q ; ) D (p) V (q ; ) (p) V (q ;0) E (42) F (p) 2 (q ; ) D (p) V (q ; ) (p) V (q ;0) (p) V (q ; ) (p) V (q;0) E (43)
Assuming that the parti les are un orrelated (whi h is true in a dilute
so-lution), we de ne a set of self-intermediate s attering fun tions for ea h typ e p
( p. eq. (21)): F (p) s (q; ) D exp iq[r m ( ) r m (0)℄ E (44)
where m is a parti le of the typ e p: The arguments of subse tion 2.4.a lead to
the following `partial' relations:
F (p) 1 (q ; )=h N p iF (p) s (q ; ) (45) F (p) 2 (q; )=hN p i 2 1+ F (p) s (q ; ) 2 (46) ( f. eqs. (26), (29)) where h N p
i is the statisti al average of the numb er of
p-parti les in the s attering volume V (here we negle t u tuations ÆN p
assuming
that the s attering volume is large enough, N p
1, whi h is equivalent to the
Gaussian approximation). The total orrelation fun tion F 1
of the multi omp onent
system then is:
F 1 (q; )= X p M 2 p D (p) V (q; ) (p) V (q;0) E = X p hN p iM 2 p F (p) s (q ; ) (47)
Turning to the total homo dyne orrelation fun tion, F 2
(q; ) / hC 4
i, we note
that it is only the terms of typ e D (p) V (p) V (p 0 ) V (p 0 ) V E that survive in F 2 after the
averaging (re alling the mutual statisti al indep enden e of the parti les and that D exp iqr E =0). Hen e
F 2 (q ; )= X p M 4 p F (p) 2 (q ; )+ X p X p 0 6=p M 2 p M 2 p 0 F (p) 1 (q ;0)F (p 0 ) 1 (q ;0)+F (p) 1 (q; )F (p 0 ) 1 (q ; )
Finally, using eqs. (45), (46), (44) we get (re alling that F (p) s (q ;0) =1): F 2 (q ; )= X p X p 0 M 2 p hN p iM 2 p 0 hN p 0 i h 1+F (p) s (q ; )F (p 0 ) s (q; ) i (48)
Using eqs. (22), (47) we get:
F 2 (q ; ) F 2 (q ;1)= X p M 2 p h N p iF (p) s (q ; ) 2 = F 1 (q ; ) 2 (49) where F 2 (q ;1)= X p M 2 p hN p i ! 2 (50)
These results show that the relationship (30) b etween F 1
and F
2
stays valid also
for p olydisp erse systems, and the same is true on erning the Siegert relation
b etween g (1)
and g (2)
1 (eq. (31)).
Let us now turn to a solution of p olydisp erse di usive parti les moving with
distin t drift velo ities v 1
,..., v p
. The rst orrelation fun tion, eq. (47), then reads
(on using eq. (37)):
F 1 (q ; )= X p hN p iM 2 p exp iqv p q 2 D p (51)
The hetero dyne DLS exp eriment (see eqs. (14), (15)) allows to obtain
< h F 1 (q; )=F 1 (q;0) i
whi h is now a sum of p damp ed os illation mo des with
frequen ies ! 1 ,...! p (! p 0 =qv p
0); f. eq. (39)). The homo dyne DLS exp eriments (see
eqs. (16), (49)) measure (g (2) (q; ) 1) = F 1 (q; ) 2 =F 2
(q;1) whi h now in ludes
os illating terms re e ting all the velo ity di eren es:
g (2) (q; ) 1 = X p 2 p exp( 2q 2 D p )+ +2 X p X p 0 >p p p 0 exp h q 2 (D p +D p 0 ) i os h q v p v p 0 i (52)
see eqs. (16), (49). Here p
is a material onstant related to the p olarizability of
p-parti les and their on entration:
p = hN p iM 2 p P p 0 hN p 0iM 2 p 0 (53)
For the ase of only two distin t typ es of parti les, the hetero dyne fun tion
< h F 1 (q; )=F 1 (q;0) i
onsists of just two os illating terms with frequen ies ! 1
and
! 2
. Con omitantly, the homo dyne fun tion g (2)
(q ; ) 1 in ludes only one os illating
!=j! 1 ! 2 j= qv ; v v 1 v 2 (54)
With no drift (for pure di usion) the DLS result, eq. (52), transforms to
q g (2) (q ; ) 1 = X p p exp( q 2 D p )= Z 1 0 G( )e d ; =q 2 D (55)
The integral expression is generally valid for systems of p olydisp erse s attering
obje ts with di erent di usion onstants D . Thereby the DLS te hnique provides
information on the size distribution of the s attering obje ts: the rate re e ts
the parti le size (hydro dynami radius) R H
( f. eq. (33)), while the fun tion
G( ) is related to p
, hen e to the partial on entrations N p
=V of the parti les
( f. eq. (53); the dep enden e R H
(M p
) is exp e ted to b e known).
To obtain G( ) based on DLS data an inverse Lapla e transform in eq. (55)
must b e done, whi h is a diÆ ult problem. It is usually solved using the sp e ial
regularization programs like the CONTIN algorithm [31,32℄. The latter pro edure
allows to distinguish b etween two parti le p opulations if their resp e tive rates
di er by a fa tor of 5 or more and the ratio of their intensities G( 1 )=G( 2 ) is b elow 10 5 .
3. Results: Unusual os illations in the homodyne DLS
orrelation fun tion of TAA solutions
TAA self-assembling nanowires is an example of a system with ompli ated
multi-level internal organization. To monitor the size distribution and the growth
of the aggregates ( brils) at various exp erimental onditions is a hallenging task.
As explained ab ove, the DLS te hnique is a suitable to ol to deal with this
problem (by getting G( ), f. eq. (55)). However, the time-dep enden e of the DLS
homo dyne orrelation fun tions turned out to b e unusual for TAA solutions: it
involves well-de ned os il l ations (with more than a dozen p erio ds) of the intensity
orrelation fun tion g (2)
(q ; ) in the long time range for all irradiated samples
of TAA solutions in hloroform. As shown b elow, these os illations annot b e
readily explained with the argument whi h led to eq. (52) due to their unusually
long-lasting hara ter (see se tion 4.1 b elow).
For the purp ose of the analysis we t the time-dep enden ies of the DLS intensity
orrelation fun tions in analogy with eq. (52). It turns out that g (2)
1 shows an
initial quasi-exp onential de ay (with a few relaxation mo des) followed by a damp ed
os illatory relaxation (with a regular p erio d and slow de ay) and by the terminal
exp onential de ay at very long times:
g (2) ( ) 1' X i G i exp( i )+G os exp( os ) os( 2 T os )+G e exp( e ) (56)
see the values of the tting parameters in Table 2. While the initial (pre-os illation)
de ay generally involves few relaxation mo des (with i
< T os
), for the terminal
de ay ( e
> T os
) just one mo de is typi ally enough for tting. The de aying
exp onents in eq. (56) orresp ond to di usion of parti les with hydro dynami radii,
R H
R H =Ksin 2 ( =2) ; K 16k B T 3 n 2 '0:062 m s (57)
( f. eqs. (3), (33), (52)). Note however, that the fastest mo des with i
<310 7
s
an b e ex luded from the onsideration of the TAA aggregation kineti s as they
formally orresp ond to R H
<0:1nm (for =90 0
), i.e., to submole ular s ales.
Our DLS exp eriments show that self-assembly (indu ed by white-light irradiation
as explained in se tion 2.1) is a prerequisite for the os illations in g (2)
. Typi ally
just a few se onds of irradiation is enough to impart the os illating hara ter of g (2)
.
For example, Fig. 3 shows the homo dyne signal from TAA solutions monitored after
short light pulses (irradiation time of t ir r
=5{15s provides subsequent TAA
dark-phase self-assembly within . 10s [14℄). The long-lasting os illations ( os =T os ' 4 for t ir r
=5s and 10s) are apparent for all urves, with a mo derate de rease
of the p erio ds T os
for longer pulses. For the shortest pulse, t ir r
=5s, the initial
de ay ontains a few mo des related to oligo-aggregates (relaxation rates in the range
2 10 6 s< i <7 10 4 s orresp ond to 0:6nm<R H <0:22m, f. eq. (57)) p ointing
to a substantial amount of small aggregates in the TAA solution. Relatively noisy
hara ter of the intensity orrelation fun tion g (2)
for <10 3
s re e ts the presen e
of tr ansient modes orresp onding to growing brils with lengths up to 200nm
whose on entration hanges during 1hour of monitoring in the DLS exp eriment.
However, for longer irradiation times (t ir r
=10 and 15s) the submi ron aggregates
are not dete ted any more (see Table 2), manifesting a signi ant growth of the
self-assembling TAA brils with the irradiation time. In parallel, some mi rometri
b ers an b e seen with the naked eye as they glitter in the laser b eam in these
solutions. Su h hanges in the aggregate sizes are in line with the bril growth
b ehavior rep orted in ref. [14℄.
Os illations with p erio ds T os
= 2=! 0
' (5:8 7:5) ms whi h are observed for
the intermediate time-s ales (T os
=4 < < e
) formally orresp ond to frequen ies
! 0 =qv ( f. eq. (54)) with v = 0 2n 1 T os sin( =2) ; 0 2n '2:210 5 m (58)
( f. eq. (3)). Hen e v ' 41 53m/s for the three urves of Fig. 3 (we used
=90 0
and assumed that qv =qv).
The presen e of os illations in the long-time range of the DLS auto orrelation
fun tion g (2)
is a well-known phenomenon in systems with strong light absorption
where the in ident laser b eam ause heating of the illuminated part of the
solution [15{ 17℄. Indeed, su h heating an ause the thermo-gravitational onve tion
in the sample. As the ow velo ity v(r ) generally dep ends on the p osition r , the
parti les in di erent parts of the s attering volume V an move with di erent drift
sp eeds v p
leading to os illations as des rib ed in se tion 2.4. ( f. eq. (52)).
The solution of self-assembled ondu tive TAA brils we study is indeed
har-a terized by strong absorption at the laser frequen y, see Fig. 1d. One ould
therefore exp e t the emergen e of a onve tive ow (due to laser heating) leading
to some os illations in g (2)
: However, the prin ipal di eren e b etween the os illations
in the present system and the previously rep orted os illations in the light-absorbing
solutions is the long-lasting hara ter of os illations in our ase with more than a
v p
(r) hange smo othly in the spa e. As a result the s attering volume V should
ontain parti les with di erent drift velo ities giving rise to a sup erp osition of
os illating terms with di erent p erio ds in g (2)
. An analysis [15,33℄ (see also
se tion 4.1 b elow for details) shows that in su h situations the amplitude of
os illations in the auto orrelation fun tion g (2)
rapidly de ay with time , in
ontradi tion to the b ehavior of our system (a omparison is presented b elow in
Fig. 12).
In order to larify the origin of the os illations we p erformed a detailed
exp erimental investigation of this phenomenon. The list of the main exp eriments
an b e found in Table 1. Most of them were done at the maximum laser p ower
P 0
=21 22mW whi h ensures the b est signal-to-noise ratios. In ea h exp eriment we
monitored the initial laser p ower P 0
(at p oint `0', see Fig. 2), the fra tion P 2
=P tol 2
of the light p ower left in the b eam after passing through the sample (at p oint `2'),
and the relative intensity of the s attered light, I 3
=I tol 3
(at p oint `3' orresp onding
to s attering at =90 0
). 3
The well-de ned os illations in the fun tion g (2)
were
dete ted at long times in all ir r adiated samples of TAA hloroform solutions for
all on entrations studied (ranging from 0.375 to 7.5mM, or 0.023-0.45 vol%). On
the ontrary, no os illations were found in non-irradiated samples. Table 1 also
shows the values of the os illation p erio d T os
( f. eq. (56)) in the ases when
os illations in g (2)
( ) were visible in time range 10 3
s< <10 1
s.
We also observed the following interesting phenomenon: as so on as the sample
was illuminated with white light for a few se onds, the laser b eam start to shift
verti ally downwards (see Figure 4), while the solution starts to visibly absorb the
laser light. Within the rst minute after swit hing-on the laser the in ident b eam
progressively de e ts downwards and often form an interferen e ring pattern [34℄
b ehind the sample. The maximal verti al shift (after stabilization of the pattern)
for the 7.5 mM solution irradiated for 1 hour orresp onds to an angle (0)
?
'0:02
for the entre of the b eam and up to (r )
?
'0:045 for the di ra tion one pattern
around the entral sp ot (see Table 1). For brie y irradiated solutions (like 5s
irradiation followed by in ubation in the `dark' when only the DLS red laser light
is applied) we dete ted a smaller verti al shift with (0)
?
'0:004 and no apparent
di ra tion pattern.
Although the app earan e of irradiated for 5s and then in ubated TAA solutions
is very similar to that of the original monomeri solution (lightly yellow and
transparent), su h solutions do absorb and s atter laser light, and the absorban e
in reases with the in ubation time t in
(see Table 1 for the transmitted light
intensity, P 2
(t in
), for 7.5mM solution in ubated in the dark for time t in
after
irradiation for 5s). By ontrast, the solutions irradiated for 1h manifest the
maximum absorban e whi h slowly de reases during the in ubation time ( ompare
7.5mM solutions with t ir r
= 1h and t in
= 0 19h). Con omitantly, they hange
olor from brightly green to brownish ( f. the absorban e plot in Fig. 1d).
While the p erio d of os illations strongly dep ends on the irradiation time (e.g.
T os
710 3
s for 7.5mM TAA solutions for t ir r =5s, and T os 2:510 3 s for 3
The intensity values P 2
and I 3
are normalized by the resp e tive intensities, P tol 2 and I tol 3 ,
measuredforpuretoluene usingthe samelaserp owerP 0
ir r
the dark (slowly in reasing with t in up to T os ' 8:510 3 s for t ir r =5s, and up to T os '3:510 3 s for t ir r =1h after t in
=19h), see Figures 5a,b and Table 1.
Moreover, the os illations b e ome faster at higher on entrations (see Figure 6).
These observations are in a ord with the exp e ted variations of the laser light
absorption: it in reases at higher on entrations and for longer irradiation time,
but b e omes weaker with longer in ubation time t in
:
Following the literature rep orts on g (2)
os illations aused by light
ab-sorban e [9,15,16,33,35℄ we studied the in uen e of the s attering angles on
the intensity orrelation fun tion g (2)
. As the os illations are observed at any
on entration, dilute TAA solutions ( =0:375mM) were used in order to minimize
the absorption of the laser b eam and hen e to weaken the dep enden e of the
s attered intensity I 3
on the p osition of a s attering element along the in ident
laser b eam. We used long-irradiated samples (t ir r
= 1hour) and maximal laser
intensity (22.2mW). To minimize the e e t of the in ubation time variation, we
onsidered just 3 di erent s attering angles ( =60 0 ; 90 0 and 120 0 ) for most of
the series, and made ea h series of measurements within 12min time-lags (three
measurements of t D LS
= 200s and 1-2mins for an adjustment of the s attering
angle ). The auto- orrelation fun tions g (2)
(q; ) for these three angles were
re orded roughly every hour during the rst 3 hours of the 'dark in ubation'
after the irradiation. For all these series, no dep enden e of the p erio d T os
on
the s attering angle was observed, while a weak but dete table hanging in the
os illation p erio d was registered over the ourse of this exp eriment ( f. Table 1).
Finally, the on lusion that T os
( ) = onst was on rmed by the fun tions g (2)
(q; )
re orded for the s attering angles b etween 30 Æ and 140 Æ (with 12 di erent ) after t in
= 3h30min-4h05min in ubation in the dark, see Figure 7 and 9 . This
is in ontrast with previous rep orts on light-absorbing assemblies and p olymers
where a lear dep enden e of the os illation p erio d on the s attering angle was
observed [15,16℄.
Given that light absorption leads to a lo al heating and onve tion, it is essential
to establish the e e t of the in ident laser light p ower P 0
. When re ording the
DLS data at a xed s attering angle ( = 90 0
) for TAA solution, = 0:375mM,
illuminated for t ir r
=1 hour and kept in the dark for t in
=5 hours prior to the
measurement, we found that a variation of the in ident laser p ower results in a
hange of the os illation p erio d (see Figure 8 and Tables 1, 2). The os illations
were shown to shift to slower times as the laser p ower de reases. At the same
time the amplitude of the os illatory mo de (with T os b etween ' 610 3 and 410 2
s) de reases making it easier to separate from the fast de ay mo des (with
i
) asso iated with di usion of short bril fragments.
The dep enden ies of the os illation p erio d T os
on various exp erimental parameters
(TAA on entration ; in ident laser p ower P 0
, white-light irradiation time t ir r
,
in ubation time t in
in the `dark', and the s attering ve tor q) are summarized in
Fig. 9.
In addition, we have examined the e e t of bril size on the p erio d T os
(whi h
is p ossibly related to sedimentation of self-assembling aggregates). To this end,
two soni ation exp eriments were p erformed. In the rst exp eriment two aged
self-assembled TAA solutions with =0:375mM and =7:5mM (whi h were prepared
by irradiation for t ir r
=1h followed by in ubation in the dark for t in
soni ated for 2 hours in the dark. The orrelation fun tions g were re orded
just b efore the soni ation (at t son
=0) and then every 30min during the soni ation
stage, and nally 3 hours after the end of soni ation (i.e., at t son +t r est with t son = 2h and t r est
=3h ). The results are shown in Fig. 10. The DLS re ording
times were hosen here to b e very short (20s) in order to avoid re-assembling of
the aggregates during the DLS measurements when the soni ation was temp orary
interrupted for instrumental reasons (on the adverse side, the shorter measurement
time is resp onsible for a higher noise level in the DLS signal). We observe that
the resulting p erio d T os
is nearly the same for all measurements.
Another soni ation exp eriment was done with freshly prepared 0.375mM TAA
solution whi h was simultaneously soni ated and irradiated with white light during
1 hour. Every 10 mins the soni ation/irradiation was interrupted for 10s for
brief DLS re ording. The results are presented in Fig. 11. The os illation p erio d
slightly de reases from T os
' 810 3
s for the rst re ord (at t son = 10min) to T os '610 3 s at t son
30min, and then it stabilizes.
4. Dis ussion
The regular os illations observed in the intensity orrelation fun tion g (2)
may
arise from the so- alled `hetero dyning' of the s attered light when the dete tor
simultaneously re eives light s attered from two (or more) distin t s attering obje ts
moving with di erent drift velo ities v 1
and v 2
, f. eqs. (52), (54). As su h obje ts
one an onsider solute parti les in separate subvolumes V 1
and V 2
(b oth b elonging
to the s attering volume) having distin t ma ros opi drift velo ities v 1
and v 2
, or
two distin t sp e ies in a ommon volume (e.g. small parti les and large aggregates)
whose motion with distin t velo ities is driven by an external for e eld.
The two p ossibilities are onsidered in referen es [15,16℄. In the rst referen e,
S haertl and Ro os prop osed an inhomogeneous onve tion pattern due to lo al
heating of disp ersed gold lusters by the in ident laser b eam: the onve tion ux
velo ity hanges from v to +v near the enter of the b eam inside the s attering
volume. In the next se tion 4.1 we shall onsider a generalization of this mo del
for an arbitrary exp erimental geometry (the shap e of the s attering volume and
the ow pattern in the solution) trying to explain the intermediate and long-time
os illations in the g (2)
fun tion.
In the se ond referen e, Sehgal and Seery intro du ed a mo del where the heavy
and the light fra tions of the solute ( omp osed of light absorbing p olyaniline or
omplexes of yto hrome and yto hrome p eroxidase) are moving with distin t
velo ities v 1
6=v 2
due to a lo al gradient of ma ros opi ow eld. It leads to the
following auto orrelation fun tion (the os illatory relaxation mo del):
g (2) (q; ) 1=A 1 exp h 2q 2 D 1 i +A 2 exp h 2q 2 D 2 i + +Aexp h q 2 (D 1 +D 2 ) i os(qv ); (59) where v =v 1 v 2 .
Let us onsider a solution with just one typ e of solute parti les hara terized by
the same di usion o eÆ ient D . In the presen e of a onve tion ow the parti les
move with di erent drift velo ities v = v (r) dep ending on the parti le p osition r .
In this ase eq. (52) transforms as:
g (2) (q ; ) 1=Aexp( 2q 2 D ) e g(q ; ); (60) e g(q; )= 1 V 2 Z V d 3 r Z V d 3 r 0 os h q(v(r ) v (r 0 )) i (61)
where we assume that the average on entration of parti les, (r ), is the same
everywhere within the s attering volume V. The os illations smear out at longer
time . Mathemati ally it is similar to the smearing of the intensity os illations in
the di ra tion pattern from a hole at large di ra tion angles [33℄. The smearing
dep ends on the geometry of the s attering volume V and the distribution of the
drift velo ities v(r ) in it.
The simplest assumption is that the drift velo ity hanges linearly in spa e,
hen e so do es qv (r) (say, it varies from a 1
to a 2
in the s attering volume). As
a result, the osine argument X, X = q (v (r) v(r 0
)), should vary within the
segment [ !; !℄, with ! =a 2
a 1
: Dep ending on the geometry of the s attering
volume V, the integration of os( X) in eq. (61) results in
e g 1 (! )= " sin(! ) ! # 2
for slit ap erture (62)
e g 2 (! )= " 2J 1 (! ) ! # 2
for ylindri al ap erture (63)
e g 3 (! )=9 " sin(! ) (! ) 3 os(! ) (! ) 2 # 2
for spheri al ap erture (64)
where J 1
is the Bessel fun tion of the rst kind and rst order. The equations
ab ove orresp ond to the three most natural geometries of the s attering volume.
The ` ylindri al' ap erture means that the volume V is ylindri al along the b eam
and that the drift velo ity v hanges in a linear fashion a ross this volume (i.e.,
in the dire tion p erp endi ular to its axis). This situation is quite natural for a
laser b eam; it was onsidered in ref. [15℄. The `spheri al' ap erture means that
the s attering volume V is restri ted equally in all three dire tions (so that V
is spheri al). By ontrast, the `slit' geometry means that all ross-se tions of the
volume V along the planes of equal velo ities (with the same value of qv (r)) have
equal areas. This situation an b e realized for a laser b eam of uniform thi kness
if the drift velo ity hanges along the b eam (but not a ross it).
The smearing fun tions, eqs. (62), (63), (64) (see Fig. 12b) are universal
fun tions of the renormalized time parameter ! . They determine the time-de ay
hara ter of the os illations in the auto orrelation fun tion g (2)
provides the fastest smearing of the os illations. Note that the full auto- orrelation
fun tion g (2)
, eq. (60), in ludes also the exp onential fa tor whose de ay is de ned
by an indep endent parameter q 2
D .
The theoreti al predi tions given in eqs. (60), (61) are ompared in Fig. 12a
with the exp erimentally obtained orrelation fun tions g (2)
. The weakest theoreti al
damping orresp onds to the absen e of di usion (D =0; g (2)
(q ; ) 1/ e
g(q; )) and
to the `slit' geometry: e g = e g 1 , eq. (62). Hen e, g (2) (q ; ) 1=A " sin(! ) ! # 2 +A 0 (65) where ! = !(q ), ! h qv (r ) i max h qv (r ) i min
(the minimal and maximal values
within the s attering volume) and A 0
is a onstant related to the ba kground
noise and slow pro esses (with slow
). However even in this ase the data
annot b e tted satisfa torily using the mo del, eq. (65): exp erimental os illations
demonstrate substantially weaker damping than any of the `di ra tion' fun tions,
eqs. (62), (63), (64). For the theoreti al fun tions, the ratios of the amplitudes
of the rst to the se ond p eak and of the rst to the fth p eak are r 12
2.9-5.9
and r 15
14-100, resp e tively, whereas the exp erimental data imply r 12
1.2-1.5
and r 15
2.2-4.8.
Su h anomalous p ersisten e of os illations in g (2)
from TAA solutions indi ates
that the observed e e t annot b e attributed to onve tion ow gradients. Rather
it may b e due to a sup erp osition of signals from two distin t s attering sour es
(either subvolumes or solute omp onents) hara terized by well-de ned but distin t
drift velo ities. In this ase we return to the os illatory relaxation mo del, eq. (59),
whi h indeed provides a go o d t of the exp erimental data ( f. Figs. 3, 12a). 4
However, it is unlikely that the s attering volume involves two subvolumes with
di erent ma ros opi velo ities, v 1
and v 2
, and that no intermediate velo ities are
present there. Thus, we are driven to onsider the last option: that there are two
distin t omponents moving with di erent velo ities in the solute. In the next
se tions we dis uss this p ossibility in detail, in parti ular we onsider the nature
of these distin t sp e ies, the origin of their drift, its dire tion and the p ossible
drift patterns in the sample.
4.2. The main prerequisites for g (2)
-os illations: sedimentation or
onve tion?
The dis ussion ab ove makes it lear that the observed os illations in g (2)
( ) are
likely to b e due to tw o sor ts of s attering parti les moving with distin t velo ities,
v 1
and v 2
. The relevant e e tive drift velo ity
v = 0 =T os (66) 4
Notethateq.(56)isequivalenttoeq.(59)aslongastheos illatorypartoftheDLS orrelation
fun tiong (2)
is related to v = v 1
v 2
. Its dep enden e on various parameters is shown in
Fig. 9. Typi ally v
'50 250 m/s in our exp eriments.
Before turning to a dis ussion on the nature of su h two sp e ies, let us onsider
the origin of their assumed oherent motion. Su h motion an arise, for example,
from sedimentation or onve tion.
In the ase of sedimentation, su h olle tive drift would b e rapid enough to b e
observable by naked eye: for a typi al DLS exp eriment lasting for t D LS
1 hour ( f.
Fig. 3 showing T os
7ms) the exp e ted parti le displa ement is v t D LS
20 m,
whi h is mu h longer than the sample size. Moreover, some TAA samples were
in ubated for many hours (see Table 1), but no sedimentation was ever dete ted
in the solutions. Furthermore, the DLS exp eriments were also p erformed on TAA
solutions soni ated in the dark for 2 hours to ut/shorten the TAA brils and
destroy their large aggregates. The soni ation therefore must diminish or suppress
the hyp otheti al sedimentation e e t: one an exp e t that the sedimentation sp eed
should de rease after soni ation. However, the soni ated samples still display the
hara teristi os illations with nearly the same p erio d (Fig. 10) or even shor ter
p erio d orresp onding to a faster v (Fig. 11).
Therefore, it is the onve tion that should b e favored as a p ossible reason
for os illations. A thermal onve tion is exp e ted here sin e the studied TAA
solutions absorb laser light rather strongly due to ondu tive (metalli ) nature of
TAA aggregates (typi ally with 20-50% of laser p ower absorb ed in the samples,
see Table 1). The light energy serves to heat the solution lo ally (around the
laser b eam) thus redu ing its density and indu ing gravitational ow as shown
s hemati ally in Fig. 13: the ow is dire ted mainly upward in the b eam region.
This onve tion ow is onsidered in more detail in the next se tion 4.3 (its
velo ity is also estimated there).
Eq. (54) shows that the frequen y of os illations, ! = q(v 1 v 2 ) , dep ends on
the s attering ve tor and velo ity orientations. For example, let us assume that
the s attering sp e ies 1 are onve ted with velo ity v 1
=v , while there is not drift
for sp e ies 2 (v 2 =0). Then v =v 1 v 2 =v , != qv = qv (67)
If the s attering ve tor q = k 2
k 1
is horizontal (whi h is the standard ase in
DLS, f. Fig. 2), the relevant velo ity omp onent is also horizontal. By symmetry
the horizontal proje tion of v is exp e ted to b e parallel to the in ident b eam (k 1 ). Hen e ! =j(k 2 k 1
)v j=kv(1 os ), so it must strongly dep end on the s attering
angle . Su h dep enden e, however, was not observed: rather, !=2=T os
is found
to b e nearly indep endent of (see Fig. 9 ). This observation suggests that the
s attering ve tor q has a p ermanent verti al omp onent q z
(indep endent of ) and
that v is nearly verti al: v=vn z
(n z
is verti al unit ve tor).
While the verti al drift velo ity in the s attering volume is pretty ompatible
with the thermal onve tion ow pattern ( f. Fig. 13 and the next se tion), it is
less lear how a nonzero q z
an emerge. Here we see 2 p ossibilities. One way to
5
The e e tive driftvelo ityv
orresp onds to nvq=k ineq.(59). As b efore, k=2n= 0
is
z
small angle b etween the dire tion of the s attered light and the true gravitation
horizontal plane. 6
Another p ossibility is related to thermal refra tion of the
main laser b eam downwards due to temp erature gradient in the verti al dire tion,
f. Fig. 4.
Let us rst onsider the p ossibility of a non-p erfe t orientation of the opti al
table (with unit normal ve tor n D LS
) with resp e t to the gravitation verti al
dire tion n z
. Indeed, due to the DLS set-up alignment pro edure, the wave-ve tor
of the primary b eam k 1
and the s attered light wave-ve tor k 2
lay p erfe tly in the
DLS set-up plane, i.e.
k 1 n D LS =k 2 n D LS =0 (68)
f. Fig. 2. Now supp ose this plane is tilted with resp e t to the horizon:
n z
n D LS
= os <1; where the tilt angle b etween n D LS
and the verti al dire tion
(unit ve tor n z
) is small: 1. Cho osing the Cartesian o ordinate system
with z-axis along n z
and y-axis p erp endi ular to b oth n z and n D LS , so that n D LS =(sin ;0; os ), we nd: != qv =j(k 2 n z k 1 n z )v z j=k v z jsin(' 1 + ) sin' 1 j (69) where ' 1
is the angle b etween the in ident b eam and the y-axis. For any value
of ' 1
the frequen y != !( ) strongly dep ends on : In parti ular, as is varied
in the exp erimental range 30 Æ
150 Æ
( f. Fig. 9 ), ! must vary at least by a
fa tor of 2. Obviously, su h strong angular dep enden e of ! ontradi ts the data
of Fig. 9 showing that T os
= 2=! is pra ti ally onstant with less than 10%
deviations. Hen e, the e e t of deviation from horizontality of the DLS setup an
b e negle ted.
The remaining se ond p ossible reason for q z
6=0 is the thermal refra tion of the
main laser b eam: the solution is strongly heated right ab ove the b eam (due to
light absorption and onve tion), while b elow it the heating e e t is mu h weaker
due to low thermal ondu tivity of the solution and signi ant upward onve tive
ow (see Fig. 13). The laser-indu ed heating thus leads to a strong refra tive index
gradient, dn=dz < 0, near the b eam. The gradient must give rise to a verti al
downward de e tion of the b eam as it passes through the sample ( f. Fig. 4),
showing the so- alled thermal lensing e e t. Su h de e tion was indeed observed in
most exp eriments on irradiated TAA solutions, f. se tion 3. The typi al de e tion
angles are b etween z
'0:004 (for brie y irradiated TAA samples, t ir r
=5s , whi h
were then in ubated for 1 hour to a omplish the brillization) and z
'0:02 for
the 7.5 mM TAA solution irradiated for 1 hour (the ase of maximum absorption).
As a result, the wave-ve tor k 1
of the primary b eam tilts down, k 1;z
= k
z
, while
the s attered light (k 2
) remains p erfe tly horizontal (due to alignment of the DLS
set-up). Hen e, q z = k 2;z k 1;z = k z
, and there is no angular dep enden e of the
os illation p erio d T os
=2=! any more:
6
TheDLSset-upisroutinely he kedforthequalityofalignment(whi hshould b eparallelto
the opti altable). However, a veri ation of horizontalityof the tableas su h is not in luded
!=v z q z =v z k z ; T os = 0 nv z z (70)
4.3. The onve tion ow pattern
Eq. (67) with verti ally oriented v gives ! =q z
v z
, hen e the dep enden e of the
os illation frequen y ! on the parameters is primarily de ned by the onve tion
velo ity v z
near the b eam. To gure out the onve tion ow eld, we adopt a
simpli ed mo del assuming an in nite sample size along the verti al axis (z-axis),
whi h is reasonable sin e the sample height (' 5 m) is mu h larger than its
radius R = 2:5mm (R is half of the diameter of the s attering ell). We also
assume that the ow is uniform along the b eam (x-axis), with 2-dimensional
velo ity in yz plane (Fig. 13). Thus, we onsider a thermo-gravitational onve tion
ow in a slit ( R < y < R) rather than in a ylindri al ell. The main goal
here is to estimate the onve tion velo ity v z
in the s attering volume, near the
origin y = z = 0 orresp onding to the enter of laser b eam. The b eam diameter
d beam
100m R. The ow eld presumably involves 2 onve tion rolls extending
up to the height H ab ove the b eam and on the distan e R b elow it ( f.
Fig. 13).
To established the onve tion ow, we need to onsider simultaneously 2 elds:
velo ity v (r ) and temp erature T(r ), or T(r ) = T(r) T out
, where T out
is the
temp erature in the bath outside the sample. The temp erature rise T is generated
by the laser light energy absorption with rate 0
I p er unit volume, where I =I(r )
is the b eam intensity and 0
is the absorption o eÆ ient. The absorb ed heat then
propagates outside the b eam by onve tion and thermal di usion a ording to the
master thermo- ondu tivity transp ort equation:
T t +vrT =D T r 2 T + 0 I=C p (71) where D T = T =C p
is the thermo-di usion onstant, T
is thermal ondu tivity of
the solution, C p
is its heat apa ity p er unit volume and =t is time-derivative
whi h is set to 0 b elow as we onsider a stationary ow. The ow velo ity is
governed by the Navier-Stokes equations with kinemati vis osity = = 0
, where
is the shear vis osity of the solution and 0 is its density: v t = r 0 + f 0 +r 2 v (72)
where is ex ess lo al pressure equal to the di eren e b etween the true pressure
and the ideal pressure in the liquid at rest without heating: = tr ue
+ 0
gz. The
volume for e f here is the buoyant for e (dire ted upwards) due to temp erature
expansion of the liquid, f =f b n z , with f b = g '{ 0 gT (73) where 0
, g is the gravity a eleration and { 1:2710 3
1=K is the thermal
0:5%), hen e it is p ossible to approximate their density, heat apa ity, thermal
ondu tivity and vis osity by the pure solvent values (at 20 Æ C ): 0 1:49 g m 3 ; C p 1:4310 7 erg m 3 K ; T 0:1310 5 erg s mK ; 0:57 P (74) Hen e D T 910 4 m 2 =s and 410 3 m 2
=s meaning that heat di usion is slow
as ompared with vorti ity di usion: D T
. In addition, we adopt the following
onditions (veri ed at the end) to simplify the mo del:
D T Rv 0 ; R 2 v 0 H (75) where v 0
is the hara teristi ow velo ity.
The buoyant for e (73) driving the ow is de ned by the temp erature eld
T(y;z) whi h is des rib ed b elow. We rst fo us on the region near the origin,
jyj R. Here the ow velo ity v must b e dire ted verti ally (along z-axis),
v v 0
(z)n z
, where the subs ript `0' indi ates that y =0. Then eq. (71) gives in
the stationary ase (T=t 0)
v 0 T z =D T r 2 T + 0 I(r ) C p (76) where r 2 T = 2 T z 2 + 2 T y 2
. It is useful to re all that the b eam intensity is lo alized
in the region r p y 2 +z 2 .d beam
: I(r ) is negligible for z d beam
. In this regime
eq. (76) is analogous to a simple di usion equation (with z=v 0
playing the role of
time). Assuming that v 0
(z) onst ( f. eq. (85) b elow) and negle ting the term
2 T=z 2 in r 2
T, the solution of eq. (76) for z d beam reads: T(y;z)' f W q 2 2 y exp y 2 2 2 y ! ; 2 y 2D T z=v 0 (77) where f W = 0 P v 0 C p (78) and P = Z I(r)dydz (79)
is the total laser p ower. Here we take into a ount that T 0 b elow the b eam
(z <0, jzj d beam
) where T T out
.
Now we note that the term 2 T=z 2 an b e negle ted in eq. (76) if y z whi h is equivalent to z D T =v 0
(we used here the solution, eq. (77)). This
ondition an b e understo o d with a simple s aling argument. For a given distan e
z from the b eam enter, the hara teristi drift time is jzj=v 0
and the
thermo-di usion time is z 2
=D T
( f. eq. (71)). Therefore the drift dominates over di usion
in z-dire tion if z z D D T =v 0 =v 0 D ; D D T =v 2 0 (80)
whi h oin ides with the stated ab ove ondition to negle t the term T=z in
eq. (76).
More general results on the temp erature eld, T(y;z), are outlined in the
App endix A. The predi tions, eqs. (77), (A9), (A11), (A12), are imp ortant. They
show that the heating e e t is lo alized ab ove the b eam in a zone jyj. y
p
zz D
whi h is narrow as long as y
R. The latter ondition is satis ed if
z H R
2 =z
D
(81)
Hen e, eqs. (77), (A9) are valid for z D
z H. For z H the laser-indu ed
heat is spread over the whole ross-se tion of the sample and get absorb ed by the
sample walls. As a result b oth T and the velo ity eld de ay exp onentially for
z H (hen e, of ourse, v 0
(z) is not onstant any more in this regime). Therefore,
the length H an b e indeed onsidered as the e e tive height of onve tion roll.
Note that H R due to the rst ondition (75). Note also that the mean
temp erature in rement at a height z ab ove the b eam, z D z H is T mean (z) 1 2R Z T(y;z)dy= 0 P 2v 0 C p R = f W 2R (82)
i.e. it do es not dep end on the o ordinate z (we used here eqs. (77), (78)).
Eq. (77) shows that the for e f b
(y;z)'{ 0
gT driving the ow ( f. eq. (73))
is lo alized near the xz plane in a signi ant region of the sample (for 0<z H,
H R). Therefore the for e f b
an b e repla ed by the surfa e for es 2F b applied at y=0 with F b = Z R 0 f b (y;z)dy
Using eq. (77) we nd that F b
do es not dep end on z for z z D : F b ' 1 2 f W{ 0 g F 0 b ; z D z H (83)
By ontrast, this for e is exp onentially small b elow the b eam, z <0, jzj z D ( f. eq. (A11)): F b 'F 0 b exp( jzj=z D ); z <0 (84)
The uniform surfa e for e 2F 0 b
ab ove the b eam applied at y=0 must naturally
generate two unidire tional simple shear ows on b oth sides of the zx plane (for
y<0 and y >0). Assuming the no-slip b oundary ondition at the solid surfa es:
v=0 at y =R, we get the velo ity pro le (v is dire ted verti ally):
v z (y;z)' F 0 b (R jyj); R<y<R; R z H (85)
The ab ove velo ity pro le (85), however, violates the volume onservation
(in om-pressibility) ondition in the onve tion slit demanding that
Z R R v z (y;z)dy =0 (86)
