Kinetics of Network Formation and Heterogeneous Dynamics of an Egg White Gel
Revealed by Coherent X-Ray Scattering
Nafisa Begam ,
1,*
Anastasia Ragulskaya,
1
Anita Girelli ,
1
Hendrik Rahmann,
2
Sivasurender Chandran ,
3
Fabian Westermeier ,
4
Mario Reiser,
2,5
Michael Sprung,
4
Fajun Zhang ,
1,†
Christian Gutt ,
2,‡
and Frank Schreiber
1,§
1
Institut für Angewandte Physik, Universität Tübingen, 72076 Tübingen, Germany
2
Department Physik, Universität Siegen, 57072 Siegen, Germany
3
Department of Physics, Indian Institute of Technology Kanpur, Kanpur, Utta r Pradesh 208016, India
4
Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany
5
European X-ray Free-Electron Laser GmbH, Holzkoppel 4, 22869 Schenefeld, Germany
(Received 23 July 2020; revised 12 November 2020; accepted 11 January 2021; published 2 March 2021)
The kinetics of heat-induced gelation and the microscopic dynamics of a hen egg white gel are probed
using x-ray photon correlation spectroscopy along with ultrasmall-angle x-ray scattering. The kinetics of
structural growth reveals a reaction-limited aggregation process with a gel fractal dimension of ≈2 and an
average network mesh size of ca. 400 nm. The dynamics probed at these length scales reveals an
exponential growth of the characteristic relaxation times followed by an intriguing steady state in
combination with a compressed exponential correlation function and a temporal heterogeneity. The degree
of heterogeneity increases with decreasing length scale. We discuss our results in the broader context of
experiments and models describing attractive colloidal gels.
DOI: 10.1103/PhysRevLett.126.098001
Egg white, apart from being a major protein source in our
daily diet, is one of the most versatile products in the food
industry due to its multifunctional properties such as
foaming, emulsifying, and gelling [1–5]. Gelation of
proteins [6], especially as a result of heat-induced denatu-
ration, leads to a three-dimensional network structure
through the formation of disulfide cross-links and hydrogen
bonds [1,7,8]. While various experiments mainly focused
on the understanding of structural evolution at the molecu-
lar level [1,9], the corresponding evolution of the micro-
scopic dynamics in a heat-treated protein gel has not been
fully explored yet.
So far, the studies of the dynamic properties of thermal
gels of proteins have only contributed to the understanding
of internal or short-time self-diffusion processes in protein
systems occurring at nanosecond and subnanosecond time-
scales [10,11]. However, the process of protein gelation
involves rather a hierarchy of length and timescales. This
requires the understanding of the structure and dynamics of
protein gels on length scales ranging from single proteins
up to the network mesh size (hundreds of nanometers to
micrometers), and timescales of milliseconds to hundreds
of seconds. This constitutes a considerable challenge both
for simulations and experiments with the need to capture
especially nonequilibrium phenomena such as aging and
dynamical heterogeneity.
Applying state-of-the-art low-dose x-ray photon correla-
tion spectroscopy (XPCS) along with ultrasmall-angle
x-ray scattering (USAXS), we follow here the simultaneous
evolution of the microscopic dynamics on length scales of
the network mesh size and the structural evolution corre-
sponding to the gelation kinetics of hen egg white. We sho w
that, under the chosen conditions, the kinetic ev olution of the
gel network can be remarkably well separated from the
dynamics. We find that the gel exhibits stress-activ ated
dynamics with an exponential rise of the relaxation time
(aging) and a subsequent steady-state ballistic motion
displaying significant temporal heterogeneity. Our experi-
ments show that the dynamical aging sets in at timescales
well beyond the formation time of the network structure. We
identify the stress-driven dynamics as dynamical rupture
ev ents which do not change the structure of the gel. The
spatial extension of these decorrelation ev ents decreases
from 100 nm to a few nm upon aging accompanied by a
lowering of the degree of dynamical heterogeneity.
The hen egg sample was purchased from a supermarket.
The egg white was extracted in the laboratory, filled in
quartz capillaries with a diameter of 1.5 mm, sealed with
parafilm, and mounted on a sample stage equipped with a
temperature-controlled heating stage (Linkam Scientific
Instruments Ltd., UK) to perform XPCS experiments in
USAXS geometry [12,13] at the beam line P10 of PETRA
III (DESY, Hamburg, Germany, with λ ¼ 1.54 Å) [14].
Applying a large beam size of 100 × 100 μm
2
allowed us
to perform the experiment at x-ray doses below the damage
threshold. We recorded a series of time-resolved scattering
patterns (with a temporal resolution of 40 ms) using an
area detector, Eiger X 4M (75 × 75 μm
2
pixel size)
mounted 21.3 m downstream of the sample (details in
the Supplemental Material [15]).
PHYSICAL REVIEW LETTERS 126, 098001 (2021)
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We first investigate the growth kinetics of the gel
aggregates by calculating the scattering invariant Q [16]
from the intensity profiles IðqÞ (see Fig. S2 in the
Supplemental Material [15]), which gives an estimation
of the flocculation parameter [17]. An exponential growth
of Q as a function of the waiting time t
w
is observed in
Fig. 1(a) suggesting a reaction-limited aggregation (RLA)
process [17–19]. A rapid increase of IðqÞ over time during
the first 160 s [see Fig. S2(a) in the Supplemental Material
[15] ] of the measurements indicates a structural evolution
[7,20,21]. During this step, major protein components
unfold and assume a different conformational state, which
facilitates an enhanced exposure of the hydrophobic and
sulfhydryl (SH) groups [7,20]. This, in turn, reduces α-
helix structures and increases β-sheet components [7,22].
The rate of increase in intensity in the USAXS profiles
reduces with time [see Figs. S2(b) and S2(c) in the
Supplemental Material [15] ] and eventually stabilizes,
showing no further structural change within the measured
q range. The stabilization of IðqÞ suggests that the major
structural evolution corresponding to the denaturation and
formation of the gel network is completed over the
measured length scales (∼60–2000 nm).
Next, we calculate the exponent δ of the power law
relation of IðqÞ as a function of qðIðqÞ ∼ q
−δ
Þ [23] in
the low q regime (0.0046 − 0.0077 nm
−1
) to determine a
characteristic parameter, the fractal dimension, which
indicates the extent of filling of the available space by
the structure in question [6,24–26]. δ as a function of t
w
is
shown in Fig. 1(b). δ gradually increases with t
w
and
eventually saturates at a value of ≈2 confirming that
the present aggregation can be described by the RLA
process [19].
We further characterize the development of the charac-
teristic length scale ξ of the gel network and determine the
mesh size of the gel in the final stages using the Ornstein-
Zernike equation (see the Supplemental Material [15])
[27,28] and shown in Fig. 1(c). The mesh size of the
network ξ (after t
w
∼ 300 s) asymptotically approaches a
value of ca. 400 nm. Note that within a few days of age of
the egg, the sample freshness does not alter our result
significantly as suggested by the SAXS measurements (see
Fig. S4 in the Supplemental Material [15]). We then
investigated the dynamics of the gel at the length scales
related to such mesh size aiming to understand its relation
with the structural evolution.
The evolution of dynamics is obtained from the two-time
correlation (TTC) defined as the covariance of the scattered
intensity [12,13,29–35]
C
I
ðq; t
1
;t
2
Þ¼
hI
p
ðq; t
1
ÞI
p
ðq; t
2
Þi
pixels
hI
p
ðq; t
1
Þi
pixels
hI
p
ðq; t
2
Þi
pixels
; ð1Þ
where I
p
is the intensity at the pixel p, and h::i
pixels
indicates an average over the pixels within a given wave
vector range q δq. t
1
and t
2
are the times at which the
intensity was measured, t
w
¼ðt
1
þ t
2
Þ=2 is the distance
along the diagonal (t
1
¼ t
2
), and the time difference
t ¼jt
1
− t
2
j is the distance away from the diagonal (see
Fig. S5 in the Supplemental Material [15]) [12,13,
29–37]. During the gelation, we performed five consecutive
XPCS runs (each run corresponding to 160 s) at different
fresh sample spots, and the corresponding TTC are
depicted in Figs. 2(a)–2(e) to follow the evolution through-
out the entire measurement time.
In Figs. 2(a)–2(c), a pronounced slowing-down of the
dynamics with t
w
is observed. Note that the dynamics is
observable only after 160 s [starting from Fig. 2(b)] when
the major part of the structural evolution has taken place
[see Figs. 1(a)–1(c)]. The inset in Fig. 2 schematically
(c)
(b)
(a)
FIG. 1. (a) Invariant as a function of t
w
calculated over the q
range of 0.003 − 0.1 nm
−1
, showing an exponential growth (red
solid line) of aggregates, (b) fractal dimension δ as a function of
t
w
obtained at the low q region of the USAXS profiles, and
(c) mesh size, as extracted from Ornstein-Zernike plo t, is shown
as a function of t
w
.
FIG. 2. TTC (C
I
− 1) collect ed at 80 °C (at q ¼ 0.01 nm
−1
)in
the time interval of (a) 0–160 s, (b) 160–320 s, (c) 320–480 s,
(d) 480–640 s, and (e) 640–800 s after reaching the temperature.
Inset schematic shows the native state of the proteins before
denaturation (0 s after heating at 80 °C) and after unfolding due to
heat denaturation (160 s after heating at 80 °C).
PHYSICAL REVIEW LETTERS 126, 098001 (2021)
098001-2
illustrates the native state of the proteins in the beginning of
the experiment (at 0 s) and the unfolded cross-linked state
after 160 s of heating that eventually forms a network [38].
At this stage, the SH groups are exposed to the exterior of
the protein molecules and are able to interact within the
molecules as well as with the neighboring molecules,
which leads to disulfide link formation [7].
The observed growth of the dynamics is suggestive of an
aging of the gel [39,40]. At a later stage [Figs. 2(d) and
2(e)], the relaxation time stabilizes around a mean value,
and interestingly, shows a periodic temporal fluctuation,
which is indicative of dynamical heterogeneity.
The relaxation times extracted from the fit of g
2
ðq; tÞ (see
Fig. S6 in the Supplemental Material [15]) as a function
of t
w
are shown in Fig. 3.AsweobserveintheTTC,the
growth of τ starts when the rapid change in structure (in
the measured q range) ends. This observation suggests that
the microscopic dynamics during the gelation process
(approximately the first 160 s after the heating started) is
too fast to be captured within our experimental time window.
Thus, the measured dynamics after 160 s of heating, when
there are no significant structural changes, could be related to
the aging or the stabilization of the networks through local
reorganization that occurs in a gel (after the gel formation).
The initial rapid increase in τ [the orange background in
Fig. 3(a)] follows an exponential growth as a function of t
w
[shown by the dashed line in Fig. 3(a)]. This behavior is
visible for all the measured q values [see Fig. 3(a)]. After
∼500 s, the dynamics stabilizes [the green background in
Fig. 3(a)] at a characteristic time fluctuating around a mean
value of a few tens of seconds [ca. τ < 100 s; see the inset of
Fig. 3(a)]. This observation seems to be unique for the
present complex system distinguishing it from conventional
colloidal or polymeric gels [41–43].
The stretching exponent γ displays a pronounced depend-
ence on the waiting time evidencing the evolution of the
dynamics during gelation. In the early times, we observe
an exponential (γ ∼ 1) correlation function. Beginning at
∼300 s the values of γ start to increase, reaching values of
more than 2 [in Fig. 3(b)] and then fluctuate well above 1
indicating a transition from an exponential to a steeper-than-
exponential dynamics [32,43].
More information about the dynamics can be obtained
from the q dependence of γ [see Figs. S8(a)–S8(d) in the
Supplemental Material [15] ] and τðqÞ [see Fig. S8(e)
in the Supplemental Material [15] ]. We observe a
strong q dependence of γ at intermediate waiting times
(∼320–640 s). Interestingly, γ shows weak or no q
dependence in the initial waiting times (before 300 s) as
well as at the later stages (after 600 s). Figure S8(e) in the
Supplemental Material [15] depicts the q dependence of
τðqÞ modeled with a power law [τðqÞ ∼ q
−α
]. The data
indicate α ≈ 1 throughout the measurement time, ruling out
Brownian diffusive motion (α ¼ 2) of the aggregates.
Instead, the data suggest that the particle displacement
increases linearly with time, which, in combination with a
compressed exponential correlation function (γ > 1), are
the signatures of intermittent hyperdiffusive dynamics
progressing by a limited number of single decorrelation
events [39,41,44,45].
The q dependence of γ can be understood in the
framework of the relaxation of stresses generated during
the network formation, i.e., during the random cross-
linking. At the early stages, during the establishment of
the gel network through cross-linking, the dynamics is
nearly exponential [see Figs. 3(a) and 3(b)] [46]. As the
sample ages, the increased cross-linking accumulates the
stresses in the gel. The relaxation of such trapped stresses
through large-scale rearrangements leads to the compressed
exponential relaxation [43,46]. We speculate that, at these
intermediate stages, the possibility of the inhomogeneous
distribution of the stress in space in the cross-linked
network leads to the q-dependent γ. As the sample ages
further, such inhomogeneity reduces, and thus, the dynam-
ics exhibits γ ≈ 2 for the entire q range.
We further explore the gel dynamics by calculating the
average spatial extension σ of the decorrelation events
adopting the method described in Refs. [47–50]. Applying
this methodology, the intermediate scattering function
fðtÞ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
½g
2
ðtÞ − 1=β
p
is expressed as a series of decorre-
lation events of the gel network
fðtÞ¼
X
∞
N¼0
P
t
ðNÞhðNÞ: ð2Þ
P
t
ðNÞ is the probability of N events occurring during
a time interval t given by a Poisson distribution
(a)
(b)
FIG. 3. (a) Evolution of relaxation time as a function of t
w
at
80 °C at different q as indicated in the legend (in nm
−1
). The inset
shows the data within the time window t
w
¼ 480–640 sona
linear scale to visualize the temporal fluctuations. The dashed
black line represents an exponential growth function. (b) The
exponent γ as a function of t
w
shows a transition from an
exponential to a compressed exponential decay.
PHYSICAL REVIEW LETTERS 126, 098001 (2021)
098001-3
[¼ðΓ
0
tÞ
N
expð−Γ
0
tÞ=N!, 1=Γ
0
being the mean time
between events], and hðNÞ is the decorrelation rate of
fðtÞ after N events of size σ given by
hðq; NÞ ≃ exp½−ðqNσÞ
2
ð3Þ
for a ballistic motion (i.e., α ¼ 1) [50]. Γ
0
is kept constant
at 0.1 and N is varied from 0 to 100. With this, we obtain
theoretical g
2
functions yielding values of γ as a function
of qσ.
The black line in Fig. 4(a) shows γ as a function of qσ
extracted from the modeled g
2
functions. The data super-
impose with the black line when the corresponding q values
of the datasets are multiplied by individual values of σðt
w
Þ
for each time interval. Figure 4(b) shows σðt
w
Þ as a function
of t
w
indicating a gradual reduction in σ with increasing t
w
.
Larger values of σ at earlier times imply that the relaxation
of the system, immediately after gel formation, progresses
through a small number of relatively large rearrangement
events. At this stage, the relatively soft and flexible
structure of the gel network facilitates such large restruc-
turing events. With increasing t
w
, the network stiffens
gradually as indicated by the increase in elastic modulus
with time in a colloidal gel [51], despite the absence of
further structural changes observed in Fig. 1. As a result of
this stiffening, the system relaxes slowly through a larger
number of small decorrelation events, which leads to a
reduced σ and an increased τ.
We observed in Figs. 2(d) and 2(e) that τ exhibits a
temporal modulation. Such fluctuations could occur due to
spatial correlations of the dynamics that restrict the number
of independent regions in the system and result in discrete
rearrangement events in the gel. These fluctuations, thus,
indicate the presence of dynamical heterogeneity in the
system [12,52]. We quantified the dynamical heterogeneity
by calculating the fourth order correlation function or the
dynamic susceptibility given by [12,13,29,39]
χ
4
¼
hC
2
I
ðq; t
1
;tÞi
t
1
− hC
I
ðq; t
1
;tÞi
2
t
1
hC
I
ðq; t
1
;t¼ 0Þi
2
t
1
: ð4Þ
Figure 5(a) shows the calculated χ
4
as a function of the
delay time. The peak position t
⋆
of χ
4
corresponds to the
characteristic time of the heterogeneous dynamics, while its
height χ
⋆
gives a measure of the degree of heterogeneity in
the dynamics. The extracted χ
⋆
and t
⋆
are summarized as a
function of q in Figs. 5(b) and 5(c). Note that the absolute
values of t
⋆
do not change significantly with t
w
.
Nevertheless, the peak height χ
⋆
significantly reduces with
t
w
. It is thus indicating that the degree of heterogeneity
reduces with t
w
[41,48], possibly through structural rear-
rangements as the elementary steps along the direction of
its equilibrium state. This is in agreement with our
observation that the size σ of the restructuring events
decreases with t
w
[Fig. 4(b)]. Consequently, the curves
of χ
⋆
as a function of qσ at different t
w
fall on the same
curve [Fig. 5(d)].
The length scale dependence of χ
4
reveals that t
⋆
decreases with increasing q as t
⋆
∼ 1=q [Fig. 5(b)] [39]
and χ
⋆
increases with q [Fig. 5(c)] [53]. The observation of
χ
⋆
increasing with q has also been observed previously
in attractive colloidal fractal gels [48]. In such type of gels,
the volume fractions can be very small compared to any
noninteracting colloidal gel [32,54]. The gelation of the
whole egg white where the total volume fraction of proteins
is rather low also supports the possibility of such type of
dilute colloidal fractal-like gel formation. The gelation of
proteins is favored when the electrostatic repulsion between
the protein molecules is substantially reduced [55].Our
results thus indicate that the electrostatic repulsive forces
between the proteins present in their native states in an egg
(b)
(a)
FIG. 4. (a) γ as a function of qσ at different t
w
(s) as indicated in
the legend. The black solid line represents the theoretical curve
used to obtain the values of σ. (b) σ as a function of t
w
showing a
reduction with waiting time.
(a)
(d)
(c)
(b)
FIG. 5. (a) χ
4
evaluated from the TTC [shown in Fig. 2(d)]asa
function of delay time, (b) t
⋆
as a function of q for two different
waiting times showing no significant change with age, and (c) χ
⋆
as a function of q showing a monotonic q dependence. Error bars
are smaller than the symbols. (d) χ
⋆
as a function of qσ showing
that the degree of heterogeneity as a function of t
w
falls on a
single curve.
PHYSICAL REVIEW LETTERS 126, 098001 (2021)
098001-4
white are reduced after heat-induced denaturation.
Eventually, the system exhibits an interesting attractive
colloidal gel-like dynamics. The observed ballistic
dynamics (τ ∼ q
−1
) with χ
⋆
∼ q is possibly due to slow
compaction of the gel through discrete rearrangements
[48]. At a smaller length scales, especially at the length
scales of the mesh size (∼400 nm and corresponding
q ¼ 0.016 nm
−1
), these events might be distinct and
observable, which results in higher χ
⋆
values at smaller
length scales (i.e., at higher q). On the other hand, while
probing at large length scales (i.e., at smaller q), the system
appears to be rather homogeneous with less temporal
fluctuation in the dynamics (i.e., smaller χ
⋆
values).
To conclude, we simultaneously investigate the kinetics
of thermal gelation and dynamics of an egg white gel by
employing state-of-the-art XPCS along with USAXS on
length scales of the network mesh size. A rapid increase in
scattering intensity at the onset of the reaction-limited
aggregation process is observed. The microscopic dynam-
ics reveals an exponential rise of the relaxation time and a
subsequent steady-state ballistic and heterogeneous
dynamics. We discuss the observed dynamics in the
framework of a stress-activated ballistic dynamics in a
low volume fraction attractive colloidal gel. Our study
paves the way for future experiments following protein
gelation and aggregation on length scales from nm to
microns shedding light on processes highly relevant for the
food industry and soft matter physics.
The authors would like to thank the DESY (Hamburg,
Germany) for beam time. The authors also thank Motiur
Rahman Khan for his suggestions. This work was
supported by the DFG and BMBF. N. B. acknowledges
the Alexander von Humboldt-Stiftung for a postdoctoral
research fellowship, A. R. acknowledges support from the
Studienstiftung des deutschen Volkes and C. G. acknowl-
edges BMBF (Grants No. 05K19PS1 and No. 05K20PSA)
for financial support.
*
nafisa.begam@uni-tuebingen.de
†
fajun.zhang@uni-tuebingen.de
‡
christian.gutt@uni-siegen.de
§
frank.schreiber@uni-tuebingen.de
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PHYSICAL REVIEW LETTERS 126, 098001 (2021)
098001-5
![FIG. 5. (a) χ4 evaluated from the TTC [shown in Fig. 2(d)] as a function of delay time, (b) t⋆ as a function of q for two different waiting times showing no significant change with age, and (c) χ⋆ as a function of q showing a monotonic q dependence. Error bars are smaller than the symbols. (d) χ⋆ as a function of qσ showing that the degree of heterogeneity as a function of tw falls on a single curve.](/figures/fig-5-a-kh4-evaluated-from-the-ttc-shown-in-fig-2-d-as-a-d44rdl9f.webp)