CHINESE JOURNAL OF PHYSICS
VOL. 36, NO. 5
OCTOBER 1998
Diffusion Analysis of Gelatin Solutions by
Photocorrelation Spectroscopy
J. S. Hwangl, Z. P. Yang’, S. B. Dai’, S. L. Tyanl, M. T. Kuol, and C. L. Lin2
‘Department of Physics, National Cheng Kung University,
Tainan, Taiwan 701, R.O.C.
2Department of Phy sits and Chemistry, Chinese Military Academy,
Fengshan, Taiwan 830, R.O.C.
(Received April 23, 1998)
The diffusion coefficients and polydispersities of polydisperse gelatin solutions
are studied by photocorrelation spectroscopy. The method of cumulant analysis is
used in analyzing the autocorrelation data. The z-average diffusion coefficients and
polydispersities of the gelatin solutions are obtained from the first and second order cumulants, respectively. Gelatin particles are freely dispersed at low concentrations, but
form polymer networks at high concentrations. The z-average free diffusion coefficient
is independent of concentration and is proportional to q2. The hydrodynamic radii and
polydispersities of the gelatin molecules at different salt concentrations and temperatures are also studied. The results lead to the conclusion that most of the aggregates
are covalently linked and that hydrogen-bonded and collagen-folded aggregates do not
make significant contributions to the overall aggregate makeup even at temperatures
as low as 20 ‘C
PACS. 83.8O.L~ - Biological materials.
PACS. 87.64.-t - Spectroscopic and microscopic techniques in biophysics.
PACS. 83.10.Nn - Polymer dynamics.
I. Introduction
Gelatin is the denatured product of collagen, the most abundant protein in the animal
world, and is a basic component of glue, photographic emulsion and food. When a gelatin
solution of high enough concentration is cooled to a certain temperature, termed “the gel
point ”, it forms gelatin gel. Collagen-gelatin and gel systems represent an important area
of study due to the biological and medical importance of collagen, as well as the great
commercial interest in gelatin and gels [l]. In previous papers [2-31, we reported the results
of our dynamic light scattering studies on collagen and gelatin gels. This paper presents
results of our photon correlation spectroscopic (PCS) studies on gelatin solutions.
I-l. Selected previous studies on the physical-chemical properties of gelatin
solution
Osmotic pressure and sedimentatin measurements have been used extensively in the
investigation of gelatin systems. Pouradier and Venet (1950) [4-51 measured the osmotic
733
@ 1998 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
734
DIFFUSION ANALYSIS OF GELATIN SOLUTIONS BY
VOL. 36
pressure of an alkali-precursor calf skin gelatin in aqueous solutions containing various
reagents. They found that the number average molecular weight of gelatin varied from
60,800 in distilled water to 66,600 in 4.0 M urea solvent, with gelatin being molecularly
dispersed in each of the solvent systems.
A number of light scattering studies have been conducted on gelatin solutions. Only
a few of these that deal with the characterization of molecularly dispersed gelatins will be
mentioned here. For a summary of gelatin studies by various light scattering techniques,
the reader is referred to Veis [l]. Weight averaged molecular weight (Mw) and the hydrodynamic radius (Rh), also known as the radius of gyration of the molecule, are the most
common characteristics obtained from light scattering studies. Boedtker and Doty [6] used
the light scattering technique in the study of the properties of an ossein gelatin and its gel.
They found the gelatin molecule to exhibit a random coil-like shape with a mean configuration comparable to those of typical synthetic polymers and, in 2M KCNS solvent, to
possess a 96,000 weight averaged molecular weight. In addition, they also found that at all
temperatures below the equilibrium melting temperature, gelatin aggregates were readily
discernible, even in extremely dilute solutions that were incapable of gelling. The aggregate
size, however, was found to be related to the temperature and thermal path by which the
gelatin solution was measured. Williams et. al. (1954) [7] obtained a 258 A weight averaged
end-to-end chain extension by assuming a random coil model in sedimentation analysis.
Gallop (1955) [8] denatured ichthyocal and conducted, in a sodium citrate buffer at
pH of 3.7, a variety of physical chemical measurements on the sedimentation coefficients, diffusion coefficients, particle specific volume, and the intrinsic viscosity of the parent gelatin.
In addition, using light scattering technique, he obtained weight averaged molecular weights
of 1.7 x lo6 and 70,000 for collagen and gelatin, respectively. Boedtker and Doty (1956)
[9] repeated Gallop’s study and obtained for collagen and gelatin, respectively, number averaged molecular weights of 310,000 and 125,000 and weight averaged molecular weights
of 345,000 and 138,000. They concluded that three molecules with substantially different
molecular weights were formed upon denaturation from each collagen molecule. Extensive
studies of sedimentation ultracentrifuge measurements [l] later clarified the different components in the parent gelatin from collagen. A single coherent picture can now be drawn
to account for the above discrepancies and will be presented in the next section.
Courts and Stainsby [lo], studying gelatin denatured from ox hide limed collagen,
found molecular weights of 640,000 to 32,000 depending on the solvent systems and gelatin
concentrations examined. They also investigated a series of acid-processed ox bone gelatin
fractions, finding weight average molecular weights of 1.94 x lo5 to 0.25 x 105. Veis et.
al. [ll-151, employing a variety of short neutral or acid extractions, studied gelatins from
unlimed purified steer hide corium collagen. Molecular weights of 8.3 x lo6 to 0.2 x lo6
were found in various fractions.
Previous research [22-231 examined gelatin solution derived from frog skin. Depending
on gelatin concentrations examined, hydrodynamic radii of 3000 to 200 A were obtained.
At constant gelatin concentration, however, the hydrodynamic radius was found to decrease
with increasing solution ionic strength.
Light scattering results indicate that although gelatins are molecularly dispersed in
aqueous solutions, they are extremely heterogeneous, possesing molecular weights that vary
over a fairly wide range up to values of approximately 106. Obviously, therefore, a single
J. S. HWANG, Z. P. YANG,
VOL. 36
735
configurational model is unable to represent all gelatin types.
I-2. The physical structure of
gelatin
The orderly hydrogen-bonded collagen monomer molecular, tropocollagen, can be
readily melted out by heating monodisperse collagen solutions to about 40 “C in acidic
solutions. This collagen to gelatin conversion is known as denaturation, and is relatively
quick, being completed within a few minutes over a small temperature interval. The resulting disordered tropocollagen molecule falls apart in one of the three ways [l]:
If there are no additional restraining bonds between chains three randomly coiled
single-stranded peptide chains called the (1~ component, each possessing sedimentation constant (S!&,) of 3.OS, are created. In those cases where two chains are covalently crosslinked, denaturation leads to two particles: one configuration of the cx chain, the other a
two-straned molecule with S&, of 4.5s caqlled the /3 component. In the final case it can
be imagined that at least two covalent cross-links hold the three chains together. This
three-chain structure is called the y component. In the denatured solution each particle
may contain from one to more than ten covalent cross-linked Q chains.
Most work on gelatin preparation has been carried out in the laboratories of glue
and gelatin manufacturers. Their aim is to maximize the amount of soluble protein with
minimal degradation of the extracted gelatin. The main problem for these manufacturers
lies in identifying and reproducing a well-defined starting material. In addition, in the
photographic industry, fine and homogeneous gelatin emulsions are essential. With this in
mind, photon correlation spectroscopy (PCS) p rovides a very powerful approach with which
to monitor new procedures designed to improve upon the aforementioned inconsistances.
II. Photon correlation spectroscopy
As photon correlation spectroscopy has been widely discussed in the literature [16211, only the most important relations pertaining to the extraction of diffusion coefficients
from experimental data will be summarized. The field correlation function, G(r)(r), and
its normalized field correlation function, g(r)(r), are defined, respectively, as:
G(r)(r) =< E;(t)E& •t r) >
g(l)(r) = G(r)(r)/G(r)(O)
where < > denotes either an ensemble or a time average; E,(t) is the electric field of
the scattered light at time t, and r is the delay time. Similarly, the intensity correlation
function, G(‘)( r ), and its normalized function, g(“l(r), are:
Gc2)(r) =< &(t)l,(t t r) >
gt21(r) = G(2)(r)/G(21(0)
w h e r e 1$(t) = E:(t),?,(t) is the intensity of the scattered light. If many independent
scatterers are present in the scattering volume, the scattered field is a Gaussian random
field, with gt2)(r) being then related to g(r)(r) by the following Siegert relation:
g(2)(r) = 1 -I-
1g(l)(r)12
736
DIFFUSION ANALYSIS OF GELATIN SOLUTIONS BY .
VOL. 36
For a dilute monodisperse solution of spherical particles, in addition to solutions of
nonspherical particles which are small compared to the reciprocal of the scattered vector Q,
the correlation function becomes:
Ig(‘)(~)l = exp(-rr)
(1)
Note that T = Dq2 where D is the translational diffusion coefficient of the particles and is
related to their hydrodynamic radius (Rh) through the Stoke-Einstein equation:
D = kT/6m/Rh
(2)
where k,T, and 77 represent, respectively, Boltzmann constant, absolute temperature, and
solvent viscosity. For spherical particles, hydrodynamic radius (Rh) equals geometric radius
(R)*
For polydisperse solutions, Eq. 1 is generalized to:
I@)(T)~ =
.lrn F(r) exp(-l3)dr
(3)
where 1” F(r)dT = 1, and F(I’)dr is the fraction of total scattered intensity of light due
to those particles possessing decay rates between r and r + dr. Following Koppel [17] and
Cummins [20], Eq. 3 is expanded as:
(4)
where (I’) = Jo” rF(I’)dT
Pn=
s
o
represents the mean decay rate;
w qr)(r- < r >ydr
is the nth moment of F(T) about the mean decay rate; and k, is the mth cumulant of F(r).
Note that
ICI =<
r >= p1
b = ~2
k3 = ~3
k4 = ~4 - 3(/J2)2
Also note that first cumulant, ICI, is directly propertional to the z-averaged diffusion coefficients, D, and equals D,q2 where q is the scattering vector.The second cumulant, k2,
is equal to the second order moment, and is a good measure of the relative distribution
width-the polydispersity. Frequently, polydispersity is estimated from k2/kt or ~2/ < I? >2.
VOL. 36
J. S. HWANG, Z. P. YANG, . .
737
III. Materials and methods
Various amounts of limed ossein gelatin solids were added to various amounts of
solvents with different ions and/or different ionic strengths (refer to the first column of
Table I) in preparing solutions of different gelatin concentrations and/or different ionic
strengths. These solutions were kept at 4 “C at least overnight allowing the solids to swell.
The solutions were then rapidly heated to 40 “C for an hour, stirred for several minutes and
then diluted to a series solutions of different concentrations. This set of dilute solutions were
loaded into centrifuge tubes and centrifuged at a speed of 48,000 g for three hours at 30 “C .
Samples were obtained by transferring, via a syringe and tygon tubing, the supernatants
of centrifuged solutions into well cleaned cuvettes. All solvents were prepared from a single
source of deionized distilled water and filtered through a 0.2~ pore size millipore filter.
Light scattering measurements were performed either on the same day as the samples
were prepared and/or after the samples had been stored in a refrigerator at 4 “C anywhere
from overnight up to several days later. Note that samples removed from the refrigerator
were heated to 40 “C for 10 minutes before any measurements were taken. Samples which
were not measured on the same day of preparation could be stored 4 “C for as long as a
week without degradation.
For a description of the correlation spectroscopy apparatus, the reader is refered to
our previous paper [a]. Prior to loading the gelatin sample into thermostatically controlled
water bath, the apparatus was calibrated at scattering angles of 31.8 and 90” with a solution
of spherical monodisperse polystyrene latex of known particle size. Correlation data of
gelation solutions were then measured at different scattering angles and temperatures. For
TABLE I. Free diffusion coefficients and hydrodynamic radii of gelatin molecules in different solvents measured at 90’ scattering angle.
Solvents
35 o
25 ’
Ingradients
Ionic
Dl
strength (10-7cm2/sec)
0.05 M NaCl
0.05 M Tris
0.15 M NaCl
0.05 M Tris
1MNaCl
0.05 M Tris
3 M NaCl
0.05 M Tris
3 M CaC12
0.05 M Tris
0.05 M
3.56 f 0.03
87
26 * 2
0.15 M
3.54 * 0.03
87
27 f 2
1M
3.22 f 0.05
88
28 4 2
3M
2.56 & 0.05
90
30 f 2
3M
2.80 * 0.03
85
31 f2
H&i)
&(r)’
0:
( 10-7cm2/sec)
(%)
Rh@)
pz/(q2
(So)
2.42 f 0.03
88
34 f 2
2.40 f 0.03
88
34 & 2
2.12 * 0.03
88
37* 2
DIFFUSION ANALYSIS OF GELATIN SOLUTIONS BY
738
VOL. 36
each sample, correlation spectra at four or five different delay times (r) were measured
such that the total delay times spanned approximately from 0.2 to 2 correlation times.
Four experiments were performed at each delay time. The experimental durations were
chosen such that the total counts of detected photopulses were higher than 106. When the
delay time per channel was set on the correlator longer than 1 psec, the correlator’s single
clip mode was used. When it was set to 1 psec or shorter, the correlator’s fast mode, which
is equivalent to the double clipped mode at zero level clipping, was preferred.
IV. Data analysis and experimental results
Since the solutions were polydisperse, the cumulant method was employed in analyzing the data. In addition, since the scattered field from dilute gelatin solutions is a Gaussian
random field, the Siegert relation can be applied. The experimentally measured intensity
correlation data, C(r), is usually represented as:
C(r;) = B + Alg(1)(r;)12
(5)
where A is a geometrical factor arising primarily from the clipping level, temporal effects
and finite coherence area of detector, and r; represents the total delay time at channel
i. Note that r; equals iAT, where i is the number of the channel, and AT is the delay
(sampling) time per channel. The background level, B, is due to the random correlation of
photopulses, and was first independently calculated from the following equation:
B = NN,/(T/AT)
(6)
where N represents the total photopulse counts; N,, the total clipped counts; and T, the
experimental duration.
The logarithms of the normalized data, Ln[(C(r) - B)/B]‘/“, corresponding to
Ln]g(‘)(r)], were first least square fitted to different order polynomials as a preliminary
check of reproducibility (ref to Fig. 1). The different order cumulants (kr, Icg, Icg, . . .) in Eq.
[4] were determined [17,20] by minimizing the following quantity:
(7)
where M corresponds to the order of polynomials to be least squares fitted; and C(r;)
represents the raw data, sum of which, is over a set of delay times (ri), with i ranging from
2 up to the last channel (Mz) where the correlation data was analyzed. Note that for a
single exponential fit, M equals 1. The proper weighting function was discussed by Koppel
[17] and chosen to be:
exp(4rlr,)
1 t exp(-2r/r,)
where r, is the correlation time being equivalent to the reciprocal of the decay rate of the
correlation function. The estimated value for r, used in Eq. [8], was obtained by analyzing
the data with W(q) = 1. The data was then reanalyzed with the r, obtained from previous
analysis (W(T;) = 1). T he values of the ICI, k2 parameters obtained from these fittings were
then plotted as a function of r,,,. The latter represents the total delay time up to the
W(r) =
J. S. HWANG, Z. P. YANG,
VOL. 36
739
D E L A Y T I M E T (BINS)
Fig. 1. Normalized correlation data from 2.0 mg/ml gelatin solution in 0.15 M NaCl and 0.05
M tris at T = 35 “C and scattering angle 19 = 90’.
Theoretical fits to
G(z+) _ B ‘I2
B
I
m=O
Dark line: first order polynomial fits (M = 1).
Light line: quadruple polynomial fits (M = 4).
channel analyzed and is equal to the bin time, AT, multiplied by &Is. Following the
procedure of Koppel and Cummins, each cumulant was then extracted from the common
intercept of the different order polynomial fit trajectories at r,,, = 0. Correlation data
obtained at 0 = 90 ” from a 2.0 mg/ml gelatin solution in 0.15 M NaCl and 0.05 M tris at
35 “C is shown in Fig. 1, along with the best single exponential fit (the first order cumulant
fit) and the quadruple cumulant fit. Values of kr from linear, quadratic polynomial fits
to correlation data with r,,, between 11 and 440 psec are plotted in Fig. 2. The mean
decay rate < F > was then extracted from common intercept at r,,, = 0 of the different
polynomial fits. We find for this particular solution
< F >= 2.10 f 0.10 x lo4 set-r
and
II, = 3.60 f 0.15 x 10e7 cmw2/ sec.
The mean hydrodynamic radius, calculated from the Stoke-Einstein relations is approximately 86 A in this particular gelatin solution. Similarly, values of k2 or p2 obtained from
quadratic and cubic fits, shown in Fig. 4, gives ~2/ < F >2 a value of 2i%, indicating moderated polydispersity. Random and systematic errors make it difficult to extract moments
higher than second order so that the result of a cumulant analysis normally only consists
of the z-average decay rate, < I? > or D2q2, and the ~21 < F >2 quantity.
VOL. 36
DIFFUSION ANALYSIS OF GELATIN SOLUTIONS BY
740
2.:
4-
7
0 = 90”
T = 35°C
Cl
0
2.11
z
z-
1.5
I.(
FIG. 2. Estimates of the first order cumulant from linear, quadratic and cubic polynomial fits
to gelatin correlation data. The extrapolation procedure of Koppel gives < Ki > or
< r >= 2.10 x 104sec-l.
e = 9on
T = 35°C
30
OA
C5
20
2'
0
a
A
I 0
“~ --------,A
C”II,C
\
QUADRATIC
-62
0
--I
500
FIG. 3. Estimates of the second order cumulant from quadratic and cubic polynomial fits to gelatin
correlation data. Extrapolation procedure of Koppel gives < Kz > or ~2 = 1.28 x 10s.
741
J. S. HWANG, Z. P. YANG, ...
VOL. 36
2.00 1
I
2
C
1
I
I
3
4
5
(mg/ml)
FIG. 4. The concentration dependence of the z-average diffusion coefficient of gelatin solutions
(Note that points with the same symbol represent results from the samples prepared at the
same time).
Each point in Figs. 4 to 7 represents a result extrapolated from a cumulant analysis
of a complete set of data of a particular solution. The concentration dependence of the
z-average diffusion coefficient and the polydispersity in gelatin solutions, at 20 “C and 35
“C, containing 0.15 M NaCl and 0.05 M Tris are shown in Figs. 4 and 5, respectively. In
both figures, note that while D, possesses an upward trend at concentrations higher than
1.8 mg/ml (the crossover concentration), the polydisperisity remains relatively unaffected.
Particles in solutions with concentration lower than the crossover concentration undergo
free diffusion motion and their z-average diffusion coefficient is called the z-average free
diffusion coefficient, Df. Similar results as those in Figs. 4 and 5 were obtained for gelatin
solutions of 1 M CaClz and 0.05 M Tris. The z-average free diffusion coefficient, Df and
the hydrodynamic radius, Rh, calculated from the Stokes-Einstein relation, along with the
polydispersity, pz/ < I >2, for gelatin solutions with different solvents, are listed in Table
I. The dependence of Df on solvent ionic strength is shown in Fig. 6. The hydrodynamic
radius and polydispersity of gelatin solutions at 35 “C increases only slightly over an ionic
strength increase of more than 30 times. When the temperature of this solution was lowered
to 20 “C, only small increases in hydrodynamic radius and polydispersity were observed.
At 20 “C, in CaC12 solvents, the hydrodynamic radius of gelatin is smaller than in NaCl
solvents of the same ionic strength. The values of the z-average free diffusion coefficient,
hydrodynamic radius, and polydispersity measured at different scattering angles for a 1.25
mg/ml gelatin solution at 35 “C are listed in Table II. The q2 dependence of Dk is shown
in Fig. 7. Note that the z-average free diffusion coefficient is directly proportional to q2.
742
DIFFUSION ANALYSIS OF GELATIN SOLUTIONS BY ‘.
4
VOL. 36
5
C (mg/ml)
FIG. 5. The concentration dependence of the polydispersity of gelatin solutions in 0.15 M NaCl
and 0.05 M tris solvent measured at scattering angle 0 = 900.
t
"E
0 3.20
';
0
FIG. 6. The ionic strength dependence of the z-average free diffusion coefficient of gelatin solution
at temperature T = 35 “C and scattering angle 0 = 90”.
V. Discussion
As particles in solution at low concentrations undergo free diffusion, the diffusion
coefficient is independent of concentration. However, at concentrations higher than the
crossover concentration, the interparticle spacing becomes comparable to particle size, particles in solution form a polymer network. The z-average diffusion coefficient, obtained from
our measurement, corresponds to the cooperative diffusion coefficient of the network and is
given by [24] D, = kT/f_%rqt where [ is th e average distance between two crosslinks or the
interchain contacts of the polymer network. As the concentration increases the number of
crosslinks also increases thus the 6 decreases and D, increases. Note that the obtained
743
J. S. HWANG, Z. P. YANG, ...
VOL. 36
_ 3.60
0
z
"E
" 3.40
:0
-
3.20
2
I
3
4'
4
( IO-@
5
6
cm-’ 1
FIG. 7. q2 dependence of the z-average free diffusion coefficient of gelatin solutions in 0.15 M NaCl
and 0.05 M tris at 35 “C.
TABLE II. Results of 1.25 mg/ml solution with 0.15M NaCl and 0.05 M tris at 35 “C and
various scattering angles.
Scattering Angles
q2(cme2)
(I)(sec-l)
D,(cmm2/ set)
p2(secm2)
21.9 o
4.27
1.26
2.96
0.84
x
x
x
x
10’
lo3
1O-7
lo6
31.8 ’
8.88
2.68
3.02
2.52
x
x
x
x
90 o
log
lo3
1O-7
lo6
5.88
2.12
3.54
1.24
x
x
x
x
lOi
lo4
1o-7
10’
crossover concentration is higher than the true value, since centrifuging samples to eliminate
dust also removes some large sample particles from solution, leading to an overestimation
of concentration.
As the hydrodynamic radius of gelatin molecules only increases a few angstroms when
solution temperature is lower from 35 to 20 “C, the majority of aggregates are most likely
covalently linked, with hydrogen bond or collagen-folded aggregates contributing only slight
to the over all aggregate make up even at relatively low temperature (20 “C). The increase in
hydrodynamic radius with increasing solution ionic strength indicates that gelatin molecules
do carry charges. At low ionic strength, the chance for charged particles to collide and form
hydrogen bonded aggregates is reduced by the repulsive Coulombic forces between charges
on the macromolecules. As solvent ionic strength is increased, however, the Coulombic
interactions are screened out more and more completely by the charges on the solvent ions,
744
DIFFUSION ANALYSIS OF GELATIN SOLUTIONS BY
VOL. 36
allowing the marcromolecules to aggregate more readily.
Boedtker and Doty failed to find any significant change in gelatin aggregates when
solution ionic strength was increases from 0.15 to 1 M and concluded that electrostatic
forces do not play an important role in the aggregates process. On the contrary, our results
indicate that gelatin molecules carry a large amount of charge so much, in fact, that the
ions in a 1 M NaCl solution are not strong enough to screen out a significant fraction of
the repulsive Coulombic forces between the gelatin molecules. In addition, our results also
indicate that a further increase in solution ionic strength effectively screens out the repulsve
Coulombic interactions, bringing about a significant increase in aggregation.
For a monodisperse sample of small particles with qL < 1, we expect either the diffusion
coefficient to be independent of q2 or the decay rate to be proportional to q2. Since our
samples are polydisperse, the z-average diffusion coefficient is proportional to q2, with the
proportionality depending on the polydispersity of the samples. This is because the smaller
the scattering angle, the more light scattering is dominated by bigger particles.
Acknowledgments
This investigation was partially supported by the Nation Science Council of the
Republic of China under contract No. NSC 87-2112-M-006-006.
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J S. HWANG, Z. P. YANG,
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