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# Nonlinear response in oligomers across chain lengths

Caroline Raepsaet, Christiane Alba-Simionesco, Cécile Wiertel-Gasquet, François Ladieu

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Caroline Raepsaet, Christiane Alba-Simionesco, Cécile Wiertel-Gasquet, François Ladieu. Nonlinear response in oligomers across chain lengths. Physical Review Research, 2025, 7 (4), pp.043244. 10.1103/35b2-ysz4. hal-05406519

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Distributed under a Creative Commons Attribution 4.0 International License
Nonlinear response in oligomers across chain lengths

Caroline Raepsaet,¹,* Christiane Alba-Simionesco,² Cécile Wiertel-Gasquet,¹ and François Ladieu¹

¹Université Paris-Saclay, SPEC, CEA, CNRS, F-91191 Gif-sur-Yvette Cedex, France

²Université Paris-Saclay, LLB, CEA, CNRS, F-91191 Gif-sur-Yvette Cedex, France

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(Received 11 July 2025; accepted 4 November 2025; published 2 December 2025)

We present third-order and first-order dielectric measurements on a series of N-propylene glycol (N-PG) oligomers, focusing on the role of molecular length N in the glass transition. To avoid polymeric effects, we restricted the study to short chains (N = 1–20). Linear dielectric susceptibility $\Delta\chi_1$ scales as $1/N$, suggesting that only the terminal hydroxyl (–OH) groups contribute significantly to the dipolar response, while the flexible backbone units remain largely inactive. Third-order susceptibilities $\chi_3^{(3)}$ exhibit a growing hump upon cooling for all N-PG, consistent with the Bouchaud-Biroli prediction. This allows us to extract the relative evolution of the number of dynamically correlated molecules $N_{corr}$, which is as expected increasing upon cooling. But surprisingly, this evolution is identical across all oligomers, regardless of chain length, even though the size of the largest molecules likely exceeds that of the dynamic heterogeneities. This robustness of the relative evolution of $N_{corr}(\tau_\alpha)$ allows to deduce that it is insensitive to the backbone linking the monomers, the fragility, the OH-bond density, or the strength of the dipole-dipole interactions among oligomers, standard criteria to describe the phenomenon. These findings challenge kinetically constrained models, which fail to account for the observed behavior, while supporting the predictions of the random first-order transition theory.

DOI: 10.1103/35b2-ysz4

I. INTRODUCTION

Despite a substantial spread of applications and the universal presence of glass in our modern environment, understanding the physics of the glass transition remains a challenge in solid-state physics and is still the subject of numerous studies [1–4] in the supercooled regime. If we cool down a glass-forming liquid, the viscosity increases, smoothly at the beginning then more and more quickly. At lower temperatures, a very small decrease of the temperature induces a huge increase of the viscosity [5,6] and of the structural relaxation time. At the glass transition temperature $T_g$, the liquid becomes so viscous that it can be considered, at the experimental timescale, as an amorphous, out of equilibrium solid: the glass. By convention, $T_g$ is defined when the relaxation time is $\tau_\alpha = 100$ s, corresponding to a viscosity $\eta = 10^{11}-10^{13}$ Pa s, depending on the value of the infinite-frequency shear modulus.

It is now well established that the relaxation of supercooled liquids is a nonhomogeneous phenomenon [5–7]. The material can be seen, at a given temperature, as a patchwork of small regions [8–10], the so-called dynamic heterogeneities (DHs). Inside a DH, the molecule dynamics are strongly correlated, but, as a first approximation, we can consider that the DHs are independent of their neighbors [6]. The total ensemble of DHs presents a wide dispersion of relaxation

times and a narrower distribution of size. Mean values of size and relaxation time increase when the temperature decreases. Direct experimental evidence and study of DH have been presented long ago for granulars [3]. But in liquids, it would require the dynamic observation of every individual molecule, which is obviously not possible.

As we cannot observe directly the DH, indirect methods have been proposed and implemented [6]. Using the principles of dielectric spectroscopy [11], when a periodic electric field $\vec{E} = E_0 \cos(\omega t)$ is applied to a polar liquid, $\omega$ being the angular frequency and $E_0$ the amplitude, the polarization $\vec{P}$ of the liquid can be written [11] as

$$ \vec{P} = \epsilon_0 \chi_1 \vec{E} + \epsilon_0 \chi_3 \vec{E}^3 + \epsilon_0 \chi_5 \vec{E}^5 + \dots, \quad (1) $$

with $\epsilon_0$ vacuum electric permittivity; $\chi_1$, $\chi_3$, and $\chi_5$ dielectric susceptibilities at the order 1, 3, and 5; and $\vec{E}$ electric excitation field. Because of the symmetry $\vec{P}(-\vec{E}) = -\vec{P}(\vec{E})$, even terms are zero.

Bouchaud and Biroli (BB) [12] predicted that (1) the third-order dielectric susceptibility $|\chi_3|$ presents a large hump at an angular frequency $\omega_p \propto 1/\tau_\alpha$ and (2) the maximum of the hump is proportional to $N_{corr}$, i.e., to the mean number of correlated particles in the DHs. The BB’s prediction can be written as follows [12,13]:

$$ \chi_3(\omega, T) \approx \frac{\epsilon_0 \Delta\chi_1^2 a^3}{k_B T} N_{corr}(T) \mathcal{H}(\omega\tau_\alpha(T)), \quad (2) $$

with $\chi_3$ dielectric susceptibility on the order of three, $a^3$ volume of a molecule, $k_B$ Boltzmann constant, T temperature of the liquid, $\omega$ angular frequency of the electric field of frequency $f = \omega/(2\pi)$, and $\tau_\alpha$ $\alpha$-relaxation time of the liquid, $\Delta\chi_1 = \chi_1(\omega \to 0) - \chi_1(\omega \to \infty)$ with $\chi_1$ linear dielectric susceptibility. $\mathcal{H}$ is a dimensionless unknown scaling

<footnote>*Contact author: caroline.raepsaet@cea.fr</footnote>

<footnote>Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.</footnote>
function, which does not depend explicitly on the temperature (only indirectly through $\tau_\alpha$), and whose modulus shows a hump at a frequency $f_p$ proportional to $f_\alpha$, with $f_\alpha = \frac{1}{2\pi\tau_\alpha}$. Note that as the $\mathcal{H}$ function is quantitatively unknown, the absolute determination of $N_{corr}$ remains extremely difficult [14]. However, the third-order dielectric susceptibility $\chi_3$ for several temperatures gives access to the evolution of $N_{corr}$ as a function of temperature [14]. Such measurements have already been made earlier for a large number of molecular systems, such as glycerol, propylene carbonate, 1-Propylene Glycol (1-PG), plastic crystals, to name a few [15–18]. Very recently a similar hump in the nonlinear shear-mechanical response was discovered [19] in two supercooled liquids (including 1-PG). This goes well with the key outcome of BB’s theoretical prediction that any nonlinear response, as soon as it probes degrees of freedom coupled to the glass transition [12], gives direct access to the relative evolution of $N_{corr}$ in temperature.

Despite these successes, BB’s prediction is a matter of ongoing discussion in the literature, in particular about the existence of a direct link between $\chi_3$ and $N_{corr}$. On the theoretical side, the use, in the seminal BB’s work [12], of a generalized fluctuation-dissipation relation was criticized and counter examples were looked after [20–22], but in the end they cannot be compared to BB’s prediction as they do not fulfill $N_{corr} \gg 1$, which is the key BB’s assumption. On the phenomenological side, two distinct mechanisms contributing to $\chi_3$ were elaborated—namely, the “absorption energy effect” [23,24] and the electrorheological effect [25]—and it was argued that they did not use any correlation length. It was however explained in Ref. [13] that the use of $N_{corr}$ was implicit in the electrorheological effect and that this effect can be viewed as a particular mechanism consistent with BB’s prediction. This allowed to physically reconcile the electrorheological effect with the two other phenomenological models—namely, the Toy model [26] and the Pragmatical model [27]—that had been elaborated independently in the mean time, and that are both explicitly consistent with BB’s prediction. In other words, this explained [13] why all these phenomenological approaches yielded the same order of magnitude for $\chi_3$. As for the “absorption energy effect,” its reconciliation with all the other models was only achieved at the mathematical level—see the end of the Appendix of Ref. [13] and/or the Sec. 4.3 of the contribution of Albert *et al.* in Ref. [28]—by assuming the proportionality between $N_{corr}$ and the quantity $N_{corr,T}$ defined below in Eq. (8). This last step is only of mathematical nature, and this is the main reason why the consensus about the interpretation of $\chi_3$ is not reached at present. A more in-depth discussion about the interpretation of $\chi_3$ is given in Sec. IV.

In the present study, we focus on a family of oligomers, N-propylene glycol (N-PG), (OH–[C₃H₆–O]ₙ-N-H), see the schematic representations of the molecules in Fig. 1, and we vary *N* from 1 to 20. Starting from a single monomer, then increasing the size of oligomer chains, the spatial oligomer extension should be of the order of magnitude of DHs around the glass transition temperature *T*<sub>g</sub>. Our main idea is to test, by dielectric linear and nonlinear (3ω) spectroscopy, whether there is, or not, an influence of the length of the molecule on the glass transition. If such an effect were to be revealed, this would give access to the size of DHs; if not, it would show

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FIG. 1. Numerical simulations for various N-PG, with $N \le 13$, of $R_{ee}$—red triangles—and of $D_{sph}$—blue dots. For each value of *N*, $R_{ee}$ is the distance between the oxygen atoms of the two ending –OH groups. The dotted red line is a guide for the eye, corresponding to a slope $\gamma_R = 0.74$. The blue dots represent the diameter $D_{sph}$ of the smallest sphere containing the molecule. The numerical uncertainties are of the order of the size of the dots. The black dotted line represents the extrapolation to our experimental range of $R_{ee}$ reported by Ishikawa *et al.* [29] by using the experiments of Allen *et al.* [30] far in the polymer regime ($N > 10^3$). Representations of the N-PG molecule have been inserted: bottom left, oxygen, carbon and hydrogen atoms are, respectively, represented as red, gray, and white spheres; bottom right, SMILES of N-PG.

that DH relaxation does not depend on the backbone linking the monomers together. Whichever of these two possibilities turns out to be true, the experimental verdict will significantly deepen our understanding of the glass transition and constrain the few first-principles theories of this phenomenon so widespread in nature.

The paper is organized as follows: We first give an overview of the experimental conditions and justify our choice of N-PG. In the second part, we show the results of our measurements, obtained from the linear and nonlinear susceptibilities. Then, we discuss our results showing that the relative increase of $N_{corr}(\tau_\alpha)$ is an extremely robust quantity since it is insensitive to many of the parameters used to characterize the various supercooled molecular systems. Finally, we critically discuss the main first-principles theories of the glass transition with respect to our data.

## II. EXPERIMENTAL SETUP AND JUSTIFICATION OF THE CHOSEN RANGE IN *N*

### A. Materials

1-PG and its oligomers (*N*-PG), HO–[C₃H₆–O]ₙ–H, form an interesting corpus [31,32]. In Fig. 1, schematic representations of the molecules have been insterted. With an –OH group at each end, the molecules are very polar (dipolar moment of 1-PG = 2.25 D) and therefore good candidates for dielectric measurements. Varying the value of *N*, we have a set of molecules with the same dielectric –OH contributors at both ends and a molecular length increasing with *N*. The series of *N*-PG that we studied are those available commercially, namely, 1-, 2-, 3-, 7-, 12-, and 20-PG: We bought 1-,
7-, and 12-PG from Sigma-Aldrich, 2- and 20-PG from Alfa Aesar, and 3-PG from ACROS ORGANICS. They were used as received.

## B. Experimental setup

The third-order dielectric susceptibility $\chi_3^{(3)}$ was measured on the dedicated setup [14,33,34] developed earlier in our group in order to separate with a very good sensitivity the small third-order $3\omega$ dielectric response from the main $1\omega$ contribution, which is many orders of magnitude higher. In this setup, two pairs of stainless steel electrodes of diameter 20 mm and separated by Mylar® spacers are immersed in the liquid of interest. They form two samples of thicknesses 13 and 26 µm measured simultaneously in the same closed cell with a finely regulated temperature. The linear susceptibility $\Delta\chi_1$ of both samples is measured in the same configuration before and after every nonlinear measurement to verify the absence of crystallization. For each oligomer, 5–7 temperatures were measured close to $T_g$, in a frequency range $10^{-1}$ to $10^3$ Hz.

When using such thin spacers, knowing precisely the actual value of the sample thickness *e* is difficult, since it may be affected by the presence of micrometric dusts that might increase the value of *e*. As this issue becomes negligible for large values of *e*, we performed additional linear measurements with samples 175 µm in thickness, on a Novocontrol Alpha-A broadband dielectric/impedance spectrometer. For each oligomer, experiments were done for 15 temperatures, in a frequency range $10^{-2}$ to $10^6$ Hz, and a sample diameter of 20 mm. Thus, we were able to remove the uncertainties on the thickness of the thin samples.

## C. Justification of the *N* range

### 1. Estimating a priori the relevant oligomer size

How can we be sure that choosing $N \le 20$ guarantees that all our *N*-PG remain in the oligomer regime, i.e., avoid all the complexities arising in polymers such as entanglement, or repletion in transient tubes? A first indication comes from the works of Gainaru *et al.* [31,32] which indicates that, in *N*-PG, these complex effects take place only for $N \ge 90$. Besides, the onset of the polymeric behavior is also commonly defined as the value of *N* for which the chain conformation follows a random walk statistics, which is the case for $N \ge 30$ according to Ref. [32].

To be fully sure that we are not in the polymer regime, we have studied the evolution of the end to end distance $R_{ee}$ in our molecules. As this is beyond any experimental sensitivity for such short molecules, molecular dynamics calculations is an efficient alternative. We used two open source codes: AmberMD [35] to prepare the input data (molecule geometry, force fields, box preparation,...) for OpenMM [36], which simulated the movement of the molecules.

In order to avoid excessively long stabilization and calculation times, the simulations have been made at high temperature, 450 K, i.e., well above $T_g$. This is substantially higher than the temperature range of our experiments, 175–220 K. However, the density weakly changes, no more than 5%–10% [37], between 450 K and $T_g$. It is therefore valid to approximate the average geometric features around $T_g$, e.g.,

the end to end distance of the molecules, by those calculated at 450 K.

For each oligomer, *N* ranging from 1 to 13, OpenMM was run to simulate the movement of 50 molecules. We check that the system has reached the equilibrium in our simulations by studying the dipole-dipole correlation function $C(t)$. Starting by design from $C(t=0)=1$, when *t* increases, each dipole decorrelates from its initial orientation; thus, $C(t)$ decreases to zero. The system is equilibrated when all the dipoles have relaxed, then we have $C(t)=0$. After equilibration, the MD calculation is extended for 4000 ps. For each time interval of 4 ps, we calculated for every molecule (1) the distance end to end, $R_{ee}$, between the oxygen atoms of the two ending -OH groups, and (2) the diameter of the smallest sphere containing the molecule $D_{sph}$. Figure 1 presents the evolution of $R_{ee}$ and $D_{sph}$, averaged on time and molecules, as a function of *N*. Both quantities follow a power law of *N*, as we detail now.

In Fig. 1, we can see that in our *N* range, $D_{sph}$ is significantly larger than $R_{ee}$. This means that, as soon as $N=1$, the molecules are not forming a straight line between both ends. Increasing *N* up to 13, from Fig. 1 one draws $R_{ee} \propto N^{\gamma_R}$, where $\gamma_R = 0.74 \pm 0.04$ is directly related to the effective dimensionality $d_{eff} \equiv 1/\gamma_R$. This shows that $R_{ee}(N)$ has not yet reached the standard value $\gamma_R = 0.5$ typical of the polymeric regime—corresponding to $d_{eff} = 2$. More quantitatively, at 320 K, one can use the viscosity and light-scattering experiments made for a number of fractionated *N*-PG samples in a theta solvent [30], which were fairly well accounted for by the calculations of conformational energies made by using classical torsional and nonbonded interaction potentials in Ref. [29]. Performed in the range $10^3 < N < 10^4$, these experiments yield $R_{ee} = \sqrt{C_\infty nl^2}$, where $C_\infty = 5.05$, $n=3N$ is the number of covalent bonds in a molecule of *N*-PG, composed of *N* monomers of three covalent bonds, and $l=0.15$ nm is the mean length of the covalent bonds in our molecules. Thus, the relationship becomes $R_{ee} = \sqrt{N} \times 0.58$ nm. Accordingly, Fig. 1 reports the result of Ishikawa *et al.* [29] as a black dotted line extrapolated down to $N=1$ to meet our experimental range. By extrapolating our numerical results slightly above $N=13$, we see that our calculations intersect Ishikawa's result for a characteristic value of *N* given by $N_{polym} \approx 15-20$. This crossover value is close to the value of 30 that we anticipated, by using Ref. [32], for the oligomer to polymeric transition. This establishes that no polymeric effects are expected in our experimental results, except for $N=20$, as shown in Sec. III D. Finally, one observes in Fig. 1 that $D_{sph} \propto N^{\gamma_D}$, with $\gamma_D < \gamma_R$ since there is a trend for $D_{sph}/R_{ee}$ to decrease slowly when *N* increases. This shows that the compactness of the *N*-PG molecules increases slowly with *N* in our selection range. This happens before the crossover that one expects for $N > N_{polym}$ toward the standard random walk characterized by $\gamma_R = 1/2$ as in the standard polymer regime of Ref. [30].

### 2. Comparison of $R_{ee}$ to $D_{corr}$

Thanks to our numerical simulations, we can translate the value of *N* in terms of typical spatial size $\mathcal{L}(N)$ of a molecule, namely, the end-to-end distance $R_{ee}$ or the diameter $D_{sph}$ of the smallest sphere embedding a molecule of *N*-PG. From
Fig. 1, we draw $\mathcal{L}(N) = qa_{\text{mono}}N^{\gamma R}$, where $a_{\text{mono}} = 0.5$ nm is the length of a monomer, $q = 0.6$ for $\mathcal{L}(N) \equiv R_{ee}(N)$ and $q = 1.0$ for $\mathcal{L}(N) \equiv D_{\text{sph}}(N)$. We neglect the tiny difference between $\gamma_R$ and $\gamma_D$. As explained in Sec. IIC 1, one has $R_{ee} \propto N^{\gamma R}$ with $\gamma_R = 0.74 \pm 0.04$. This corresponds to an effective dimensionality $d_{\text{eff}} \equiv 1/\gamma_R$, yielding $d_{\text{eff}} \simeq 1.35$; i.e., our oligomers are more tortuous than just a straight line (where one would have $d_{\text{eff}} = 1$) and much less compact than spherical objects (where one would have $d_{\text{eff}} = 3$). This contrasts with the shape of DH's which, close to $T_g$, have been evidenced in Ref. [38] to be compact, i.e., to have an effective dimensionality of 3. As a result, for a given $T$, there is an interesting crossover $N^*$, where $\mathcal{L}$ equals the diameter $D_{\text{corr}}$ of a typical dynamical heterogeneity. As $D_{\text{corr}}$ slowly increases upon cooling, this crossover $N^*$ depends on $T$, or equivalently on $\tau_\alpha$. The physical meaning of this crossover is the following: for $N < N^*(\tau_\alpha)$ most of the N-PG molecules belong to only one DH, while whenever $N \ge N^*(\tau_\alpha)$ any given molecule of N-PG spans over more than one DH.

To get an expression of $N^*(\tau_\alpha)$, we write $N_{\text{corr}} = \phi \times (D_{\text{corr}}/a_{\text{mono}})^3$ where $\phi \simeq 0.74$ is the maximal filling factor for assembly of 3d spheres. This implies that when discussing Eq. (3), the value of $N_{\text{corr}}$ is given in terms of $a_{\text{mono}}$, which in the case of oligomers represents a monomer, but in the case of a short molecule—such as glycerol—represents the whole molecule. As we shall see below, the experimental data about the absolute value of $N_{\text{corr}}$ are broadly dispersed, but as they are given at $T_g$, i.e., for $\tau_\alpha = 100$ s, we take into account, see Sec. III F, that $N_{\text{corr}}(\tau_\alpha = 100 \text{ ms}) \simeq 0.6 \times N_{\text{corr}}(\tau_\alpha = 100 \text{ s})$ to get the expression $N^*(10 \text{ ms})$ corresponding to the center of our range in $\tau_\alpha$ of our 3$\omega$ experiments. Equating $\mathcal{L}(N^*)$ with $D_{\text{corr}}$ at $\tau_\alpha = 10 \text{ ms}$, we get

$$N^*(\tau_\alpha = 10 \text{ ms}) = \left[ \frac{0.6}{\phi q^3} N_{\text{corr}}(\tau_\alpha = 100 \text{ s}) \right]^{\frac{1}{3\gamma_R}} \\ = (0.9 \text{ to } 1.8) \times [N_{\text{corr}}(\tau_\alpha = 100 \text{ s})]^{0.45} \quad (3)$$

where we have obtained the last equality by using the above-mentioned values for $\phi$, $\gamma_R$ and where the two values of the prefactor correspond, respectively, to $q = 1$ and to $q = 0.6$.

We now come to the values of $N_{\text{corr}}(\tau_\alpha = 100 \text{ s})$ reported in the literature, which appear quite dispersed, for the reasons that we shortly discuss now:

(1) $N_{\text{corr}}(\tau_\alpha = 100 \text{ s})$ might strongly depend on the considered glassformer: for example, by using the 4D-nuclear magnetic resonance technique [39,40], one finds that a typical DH contains 10 molecules for glycerol and 100 monomers for polyvinyl acetate.

(2) for a given glassformer, such as glycerol, the value deduced for $N_{\text{corr}}(\tau_\alpha = 100 \text{ s})$ strongly depends on the kind of experiment [39–41]: By using, e.g., the heat capacity spectroscopy data by extrapolating the Donth's group model, one finds from Ref. [39] that 100 molecules of glycerol are contained in a typical DH at $T_g$, different from the value of 10 obtained with nuclear magnetic resonance [39,40].

(3) the random first-order transition theory (RFOT) estimate [3] is $N_{\text{corr}}(\tau_\alpha = 100 \text{ s}) \simeq 43$ for glycerol, which has a molecular fragility, a molecular weight and a dielectric susceptibility $\Delta\chi_1$ close to those of 1-PG. Of course, the

OH-bond density is clearly different in 1-PG and in glycerol, but since we shall show in Sec. IV A 3 the irrelevance of this density the comparison of those two liquids is still justified to get an estimate of $N_{\text{corr}}(\tau_\alpha = 100 \text{ s})$.

(4) To our knowledge, the largest value ever reported in the literature comes from the field of polymers [42,43] stating that typically 1000 monomers lie within a DH at $T_g$. However this extremely large value may be obscured by the quite involved microscopic phenomena that may arise in polymers, due to the complexity inherent to these so long molecules [44].

Consider first the latter value $N_{\text{corr}}(\tau_\alpha = 100 \text{ s}) = 1000$ inferred from polymer studies. By choosing the smallest possible value $q = 0.6$, i.e., by comparing $R_{ee}$ and $D_{\text{corr}}$, we obtain the top extreme value $N^*(\tau_\alpha = 10 \text{ ms}) \simeq 41$, which is clearly longer than our longest oligomer ($N = 20$). By comparing $D_{\text{sph}}$ and $D_{\text{corr}}$, i.e., by using $q = 1.0$, one gets $N^*(\tau_\alpha = 10 \text{ ms}) \simeq 20.5$, which (nearly) corresponds to our longest oligomer.

Moving to the former cases where the estimates of $N_{\text{corr}}$ are taken from studies of small molecules, we get $2.6 < N^*(\tau_\alpha = 10 \text{ ms}) < 15$ depending on the value of $q$ and $N_{\text{corr}}(\tau_\alpha = 100 \text{ s})$ one considers in the above-discussed interval—in Sec. IV B we shall determine $N^*$ within RFOT. Overall, the fact that, apart from the top extreme case, $N^*$ lies within our experimental range, or hardly above it, justifies our choice of our $N$ range. As explained in the introduction, inspecting whether or not there is some $N$ dependence for the relative evolution of $N_{\text{corr}}$ in $\tau_\alpha$ will either reveal the absolute size of the DH's close to $T_g$, or show that the oligomer backbone does not have any effect on the growth of the DH's size in $\tau_\alpha$.

### III. RESULTS

#### A. Linear measurements

The results of the linear dielectric measurements are presented for 3-PG in Fig. 2. We obtained similar qualitative results for the other oligomers. The range of temperatures has been chosen to be somewhat larger than the one over which $\chi_3$ data are available, i.e., temperatures ranging from 196 to 220 K.

As expected, the loss contribution $\epsilon''(f)$ shows a large hump corresponding to the $\alpha$-relaxation process, Fig. 2(a), whose maximum decreases slightly and shifts to higher frequency when the temperature increases. We define $f_\alpha = \frac{1}{2\pi\tau_\alpha}$ as the frequency corresponding to the peak, as shown in Fig. 2(a), an arrow pointing out $f_\alpha$ for $T = 196$ K.

The dielectric constant $\epsilon'(f)$, Fig. 2(b), presents a plateau at low frequency. When the temperature increases, the plateau extends to higher frequencies and its height lowers. Then, it decreases toward $\epsilon'(\infty)$, independent of $T$, measured at lower temperatures, nonrepresented in Fig. 2.

We use the phenomenological formula of Vogel-Fulcher-Tammann [45–47] (VFT) to fit the temperature dependence of the $\alpha$-relaxation time:

$$\langle \tau_{\alpha} \rangle = \tau_0 \exp \left( \frac{DT_0}{T - T_0} \right), \quad (4)$$

where $T_0$ is the Vogel-Fulcher-Tammann temperature at which $\tau_\alpha$ is supposed to diverge, $D$ is the strength parameter, and $\tau_0$ is the relaxation time at high temperature.
<figure><img src="image_5.png" /><figcaption>FIG. 2. Imaginary and real parts of the dielectric permittivity of 3-PG, measured on the Novocontrol spectrometer for temperatures ranging from 196 to 220 K: (a) $\epsilon''$ and (b) $\epsilon'$ as a function of f (Hz).</figcaption></figure>

We calculated also the fragility, commonly defined by [6]

$$m = \left. \frac{d(\log_{10} \tau)}{d(T_g/T)} \right|_{T_g} \qquad (5)$$

Table I summarizes the $\alpha$-relaxation parameters of the VFT fit $D$, $T_0$, $-\log_{10}(\tau_0)$, the glass transition temperature $T_g$ corresponding to $\tau_\alpha = 100$ s, and the fragility $m$. Overall our parameters are consistent with the previous studies existing in the literature such as in Refs. [31,32,48–50]. All this parameters present significant variations for the lower values of $N$, namely, 1–3, and very weak variations above. This smooth dependence on the chain length is also observed in the case of very flexible polymer chains, such as polyisobutylene, with a nonfragile behavior as $T_g$ is approached [51].

<table><caption>TABLE I. Relaxation parameters of N-PG.</caption><thead><tr><th>N</th><th>D</th><th>T<sub>0</sub><br>(K)</th><th>-log<sub>10</sub>(τ<sub>0</sub>[s])</th><th>T<sub>g</sub><br>(K)</th><th>m</th></tr></thead><tbody><tr><td>1</td><td>16.8</td><td>115</td><td>13.9</td><td>168</td><td>51</td></tr><tr><td>2</td><td>10.9</td><td>150</td><td>13.8</td><td>195</td><td>69</td></tr><tr><td>3</td><td>9.9</td><td>150</td><td>13.7</td><td>191</td><td>73</td></tr><tr><td>7</td><td>8.1</td><td>160</td><td>13.3</td><td>197</td><td>82</td></tr><tr><td>12</td><td>8.7</td><td>160</td><td>13.9</td><td>198</td><td>82</td></tr><tr><td>20</td><td>9.1</td><td>160</td><td>14.3</td><td>199</td><td>84</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>FIG. 3. Linear static susceptibility $\Delta\chi_1$ vs $N$, number of monomer units in the oligomer. The red dashed straight line has a slope -1 and is a guide for the eye. Blue dots are experimental results at temperatures corresponding to $f_\alpha \approx 10$ Hz ($\tau_\alpha \approx 15$ ms). The relative uncertainty on $\Delta\chi_1$ is on the order of the size of the points.</figcaption></figure>

## B. Static linear dielectric susceptibility

As $\epsilon = \chi_1 + 1$, the linear dielectric susceptibility $\Delta\chi_1$ is equal to the amplitude of the sample permittivity between $f \to 0$ and $f \to \infty$ [see Fig. 2(b)]:

$$\Delta\chi_1(T) = \Delta\epsilon(T) = \epsilon(f \to 0, T) - \epsilon(f \to \infty). \quad (6)$$

Figure 3 presents a log-log plot of the linear susceptibility $\Delta\chi_1$ measured for 1-, 2-, 3-, 7-, 12-, and 20-PG as a function of $N$, the number of monomers in the oligomer, at temperatures chosen to correspond to $f_\alpha \approx 10$ Hz ($\tau_\alpha \approx 15$ ms) (similar results have been obtained at other frequencies). The blue dots are the experimental points. A red dashed line of slope -1 has been drawn on as a guide for the eye: It shows that the experimental behavior is very close to $\Delta\chi_1 \propto 1/N$. Theoretically, $\Delta\chi_1$ is given [11] by

$$\Delta\chi_1 = \frac{\mu^2(N)g_K(N)}{3\epsilon_0Na_{\text{mono}}^3k_B T}, \qquad (7)$$

with $\bar{\mu}(N)$ the molecular dipole vector of N-PG and $g_K(N)$ the molecular Kirkwood factor expressing the static average correlations between dipoles. In Eq. (7), we have used the fact that the molecular oligomer volume is extremely close to $Na_{\text{mono}}^3$ since the density—at room temperature—is mainly independent of $N$. As we shall show below—see Sec. III D—it may happen that $g_K(N)$ decreases as $N$ increases, from $g_K(N=1) \simeq 2.85$ to $g_K(N=20) \simeq 1$. However, this is a much smaller variation than the 15-fold effect observed in Fig. 3. Taking into account these estimates of $g_K$ gives $\mu(N=20) \simeq 1.69\mu(N=1)$, whereas assuming $g_K$ being constant in $N$ would yield $\mu(N=20) = \mu(N=1)$. Anyhow, $\mu(N=20)$ and $\mu(N=1)$ are surprisingly close one to another, as if the main contribution to the linear susceptibility would come from the two –OH end groups, while the other chemical units of the chain have no significant influence on $\Delta\chi_1$. This is a counterintuitive result, because the chemical units situated along the chain between the –OH groups have dipoles that are not that smaller than the –OH ending ones. On the other hand,
<figure><img src="image_7.png" /><figcaption>FIG. 4. Modulus of the 2-PG third-order susceptibility, $|χ_3^{(3)}|$ vs $f/f_α$, where $f_α$ is the α-relaxation frequency (see text). The maximum of the peak is decreasing when the temperature increases. In the insert, we present the phase of $χ_3^{(3)}$ (deg), which turns out to collapse into a temperature independent master curve. The relative uncertainty on $χ_3^{(3)}$ is on the order of the size of the points.</figcaption></figure>

this suggests that in this system, changing the length of the molecule does not perturb the dipoles that dominate the dielectric linear response. This may be viewed as an unexpected asset of the N-PG series that may simplify the analysis of the n-linear response, to which we move now.

### C. Third-order susceptibility

The modulus of the third-order dielectric susceptibility at $3ω$ is presented in Fig. 4 for 2-PG as a function of $f/f_α$, with the usual definition of the relaxation frequency $f_α \equiv \frac{1}{2\pi\tau_α}$. Similar results were obtained for the other oligomers. The shape of the curves in frequency, as well as their evolution with the temperature, are in good agreement with Bouchaud-Biroli's predictions made in Refs. [12,38] and further commented in Refs. [33,52]:

(1) For any given temperature, $|χ_3^{(3)}(f/f_α)|$ shows a large hump around $f_p \approx 0.2f_α$, which can be understood as follows. At low frequency $f < f_p$, the probing time $t_{pr} \propto 1/ω$ associated with the electric field is much longer than $τ_α$. Thus, the system significantly relaxes during $t_{pr}$ and flows to other configurations: As a result, $χ_3^{(3)}$ will decrease to the liquid trivial contribution corresponding to effectively independent molecules. In contrast, at high frequency $f \gg f_p$, the probing time $t_{pr}$ is much smaller than $τ_α$, and the system typically does not have enough time to relax, except in the fastest DHs, which become fewer and fewer as frequency increases. Hence in this solid like situation, $χ_3^{(3)}$ decreases when $f$ increases. Between these two extremities, i.e., whenever $t_{pr} \approx τ_α$ we have an important dielectric response with a maximum for $f = f_p$. Given the spread of the $χ_3^{(3)}$ data, the value of $f_p$ was obtained using a polynomial fit of the experimental curve around the peak. The resulting uncertainty interval is $f_p \times (1.0 \pm 0.073)$.

(2) The maximum of the peak is decreasing when the temperature increases: We will see in Sec. III F that from this behavior in temperature, one can deduce by which amount the

number $N_{corr}$ of dynamically correlated molecules decreases when heating up the sample.

(3) In the insert of Fig. 4, we see that as predicted by Eq. (2), the phase of $χ_3^{(3)}$ collapses onto a master curve in $(f/f_α)$.

As further discussed below, all the features of Fig. 4 are consistent with those reported on simple glass forming liquids, such as in Refs. [14–16,33,54,55].

### D. N dependence of the first- and third-order spectra

In Fig. 5, we compare the shape in frequency of the linear and nonlinear responses of the various N-PG. We have, for each value of N, chosen a temperature for which $f_α \approx 10$ Hz, so that the α relaxations are comparable. Figure 5(a) shows a broadening of the peak of $χ_1''$ when N increases. To highlight this phenomenon, we show in the lower part of Fig. 5(a) the log-log slope of $χ_1''$ in $f/f_α$, slope = $\frac{\partial \log(|χ_1''|/|χ_1'|_p)}{\partial \log f/f_α}$. This broadening of the peak of $χ_1''$ with N deserves some comments:

(1) On the left part of the peak, two distinct frequency dependencies can be distinguished. In the first group, $N = 1–7$, the shape of the peak is that of short molecular glass formers: Increasing frequency from the lowest values of $f/f_α$, one first encounters the regime dominated by the dc conductivity $χ_1'' \propto f/f_p$, before the dipoles take the leading role, yielding $χ_1'' \propto f/f_α$ as expected from the fluctuation-dissipation theorem. This standard behavior progressively changes when increasing N, the broadening of the peak being perceptible for $N = 12$ and much more visible for $N = 20$. As explained in Ref. [32], this deformation is due to the emergence of a normal mode, i.e., to the dielectric contribution coming from the dielectric dipole $\mu_{chain}$ summed up all along the chain of each N-PG molecule. This normal mode contribution appears and grows with N and our results depicted above are in good agreement with Refs. [32,48], which, for $T \ge T_g$, reported a normal mode deformation of the α-relaxation peak for $N \ge 17$.

(2) On the right side of the peak of $χ_1''$, the absolute value of the logarithmic slope decreases with N, especially between 1-PG and the other N-PG. On general grounds, this amounts to an increase with N of the width of the distribution of the relaxation times [11], which is not surprising owing to the increase of the complexity of the system with N. Another complementary interpretation, which is more specific to the dielectric linear susceptibility, can be found in the recent works of Blochowitz's group [53] showing that the larger the Kirkwood factor $g_K$, the steeper the slope $-\beta_{diel}$ of $χ_1''$ on the right flank of the α peak (see Fig. 5 for a precise definition of $\beta_{diel}$ commonly associated with the streching of the relaxation). This comes from the fact that dielectric responses are generically made of a “self” and a “cross” part reflecting, respectively, the correlation of a given dipole with itself and with other dipoles [53,56]. References [53,56] have shown that the cross part increases in magnitude with the Kirkwood correlation factor $g_K$. Hence, there is a systematic $\beta_{diel}$ variation in $g_K$ since the dielectric cross part is steeper in frequency than the self part. For $N = 1$, Ref. [53] gives $g_K \approx 2.85 \pm 0.15$ and $\beta_{diel} \approx 0.67 \pm 0.02$, which is consistent with our findings in Fig. 5(a). In Fig. 5(a), we observe that $\beta_{diel}$ decreases when N increases, converging to
<figure><img src="image_8.png" /><figcaption>FIG. 5. Comparison of the linear and nonlinear measurements for the different values of N. Temperatures have been chosen so that $f_α \approx 10$ Hz (see Table II). (a) The upper graph represents—right axis—the linear dielectric loss $\chi''_1$ normalized to its peak. The lower part—left axis—is the log-log slope of the upper curves, i.e., the logarithmic derivative of $\chi''_1$ with respect to $\log f/f_α$, slope = $\frac{\partial \log(|\chi''_1|/|\chi''_1|_p)}{\partial \log f/f_α}$. For the high-frequency flank of the $\alpha$ peak, i.e., for frequencies $f > f_α$, the absolute value of this slope reaches its maximum—called $\beta_{diel}$ in the main text [53]—for $f \approx 20f_α$. At high frequency, the shape of the curves and their discrepancies are difficult to interpret due to the potential contribution of $\beta$ or secondary relaxation modes. As they have no influence on our results, concerning lower frequencies, we will not discuss them. (b) Module of the third-order dielectric susceptibility at $3\omega$, $|\chi_3^{(3)}(f/f_p)|$, normalized by the peak value—right axis. In the insert, we compare the evolution of the FWHM for a Debye relaxation, $\chi''_1$ and $\chi_3^{(3)}$—left axis.</figcaption></figure>

<table><caption>TABLE II. Reference temperatures used to calculate the relative evolution of $N_{corr}$, with the corresponding values of $f_α$ and $\tau_α$ within the same decade.</caption><thead><tr><th>N</th><th>T<sub>ref</sub><br>(K)</th><th>f<sub>α</sub><br>(Hz)</th><th>τ<sub>α</sub><br>(ms)</th></tr></thead><tbody><tr><td>1</td><td>183</td><td>11</td><td>15</td></tr><tr><td>2</td><td>208</td><td>8</td><td>21</td></tr><tr><td>3</td><td>203</td><td>15</td><td>11</td></tr><tr><td>7</td><td>208</td><td>13</td><td>12</td></tr><tr><td>12</td><td>207</td><td>8</td><td>21</td></tr><tr><td>20</td><td>208</td><td>6</td><td>25</td></tr></tbody></table>

$\beta_{diel} \approx 0.48 \pm 0.05$ for our largest N's. According to Fig. 1 of Ref. [53], this would correspond to a decrease of $g_K$ to values close to 1 corresponding to mainly noninteracting dipoles at large N. For $N > 1$, we cannot really derive the value of $g_K$ as we do not precisely know which dipoles are relevant for $\Delta\chi_1$ and which ones self average to zero. However, in the scenario above mentioned where the only relevant dipoles are the OH ones, one would easily anticipate that $g_K \to 1$ at large N, since the probability of having neighboring OH dipoles tends to zero for our longest oligomers.

Figure 5(b) shows the module of the third-order dielectric susceptibility at $3\omega$, $|\chi_3^{(3)}(f/f_p)|$, normalized by the peak value. Overall, we notice very similar behaviors for $\chi''_1$ and $|\chi_3^{(3)}|$: broadening of the hump when N increases, 1-PG curve separated from the other ones. A global analysis of the right side of the peak in Fig. 5(b) shows the same trends as for Fig. 5(a), namely, that the 1-PG curve is the narrowest, the 20-PG is the widest. This is understandable because the arguments given above for the linear response still hold qualitatively for the cubic response (for the self and cross parts in cubic responses, see the explicit simulations of Ref. [57]). Finally, the only difference between first- and third-order spectra above $f_α$ is that the intermediate values of N are more closely packed for $|\chi_3^{(3)}|$ than for $\chi''_1$.

A more local analysis close to the maximum of the peaks can be done by computing the so called full width at half maximum (FWHM) and compare it to the case of a pure Debye system where one has FWHM$_{Debye} = 1.14$ decades. For $\chi''_1$, it is calculated directly from the experimental data. As $|\chi_3^{(3)}|$ is very asymmetric and does not go below 0.5 on the left, we consider only the right part of the hump, and multiply the result by a factor of two, to allow a quantitative comparison with $\chi''_1$. In the insert of Fig. 5(b), we can clearly confirm the similarity of behavior for $\chi''_1$ and $|\chi_3^{(3)}|$. 1-PG has the lowest values of FWHM. There is a jump for $N = 2$ and no noticeable change for $N > 2$.

Finally, the noticeable difference between the first- and third-order spectra lies in the left part of the peak, where we do not detect any sign of the normal mode for the third-order susceptibility, even when inspecting directly the behavior of the imaginary part of the cubic response (not shown). We speculate that this may come from the fact that if the relaxation of a given system is governed by two modes “a” and “b,” of respective linear dielectric strength $\Delta\chi_a$ and $\Delta\chi_b$, the relative influences of these two modes is governed by the ratio $\Delta\chi_a/\Delta\chi_b$ for the linear response, but by the square of this ratio for the third-order response. This statement is illustrated by, e.g., Eq. (2). Hence, the small normal mode hardly visible in Fig. 5(a) is not visible on $\chi_3^{(3)}$.

### E. Influence of the normal and $\beta$ modes on the determination of $f_α$, FWHM, and $\beta_{diel}$

We focus our study on the $\alpha$-relaxation mode. However, higher ($\beta$ mode,...) and lower (normal mode) frequency relaxation modes are present for the N-PG. In order to see their influence on $f_α$, FWHM and $\beta_{diel}$, the loss experimental spectra have been fitted using three peaks, corresponding to the $\alpha$, normal, and $\beta$ modes. In this paragraph, we will refer to
the “total peak” or “total contribution” as the sum of the three HN relaxation peaks: α, β, and normal.

The β mode presents a peak at frequencies higher than our measurement range. It adds a very low contribution to the right wing of the main peak and its influence on fα, FWHM, and βdiel is fully negligible for all the N-PG measured.

Concerning the normal mode, we compared the fα, FWHM, and βdiel of the total and α loss contribution for 20-PG obtained from the maximum of the total and α peaks.

(1) The comparison of the values of fα shows no difference for T > 218 K, the normal mode being very small compared to the α mode. For lower values of T, a difference gradually appears upon cooling for the temperatures in the range 204–218 K. However, the amplitude of this shift remains small; i.e., in the worst case it is not larger than the 0.5 K error bar given for Tg see Sec. IV A 1.

(2) The values of the FWHM from the total and α peaks differ from less than 5%.

(3) The relative difference between the values of βdiel from the total and α contributions is less than 0.3%.

(4) In Fig. 5(a), the left wing of the loss peak and moreover the derivative of the loss peak clearly show the emergence of the normal mode for N= 20. For N = 12, it was not reasonably possible to introduce a normal peak as the left part of the loss signal was well fitted by a single α contribution.

From these considerations, we conclude that in our experimental conditions, there is no influence of the normal mode on the determination of fα, FWHM, and βdiel for N-PG with N < 20. For 20-PG, the normal mode is clearly visible but induces modifications that are small enough to be neglected. This legitimates the calculation of these three parameters directly from of the measured global peak.

### F. Relative evolution of Ncorr in τα

Equation (2) shows [12,13] that the temperature dependence of the mean number of molecules in a DH, Ncorr, can be extracted from that of χ(3), provided that we move to the dimensionless cubic susceptibility X(3) defined as

$$X_3^{(3)}(\omega, T) \equiv \frac{\chi_3^{(3)}(\omega, T) k_B T}{\epsilon_0 \Delta \chi_1^2 a^3}, \\ X_3^{(3)}(\omega, T) = N_{\text{corr}}(T) \mathcal{H}(\omega \tau_{\alpha}(T)) + X_{3,\text{triv}}^{(3)}(\omega \tau_{\alpha}(T)), \quad (8)$$

where the definition of X(3) shows that its spectrum at each given T can be deduced from that of χ(3) by using only experimentally well-known quantities, and where the last equality arises only when assuming that the Bouchaud-Biroli’s prediction holds. X(3) is the trivial contribution that would exist even for independent dipoles and which is known from theoretical literature (see Refs. [16,33,38,58,59]). Importantly, the scaling function H(ωτα(T)) is predicted not to depend explicitly on T, but may depend on the considered liquid, and thus may depend on N. To circumvent this issue of the unknown scaling function, we consider a given value u0 = ωτα and normalize the value obtained for |X(3)(u0) - X(3)(u0)| at any temperature by its corresponding value at a reference temperature Tref. As a result, the unknown value of |H(u0)| simplifies and one obtains directly Ncorr(T)/Ncorr(Tref). In practice, we consider the peak of |X(3)|, for which u0 = ωpτα = 2πfρτα,

<figure><img src="image_9.png" /><figcaption>FIG. 6. For all the measured oligomers N-PG (N = 1, 2, 3, 7, 12, 20). (a) Relative evolution of the mean number of correlated elements Ncorr in a dynamic heterogeneity as a function of the relaxation time τα. Both Ncorr and τα are normalized by their values at a reference temperature Tref, in order to get rid of the contribution of the unknown H function. (b) Same kind of plot as in panel (a) but for Tχ(T) normalized to its value at Tref to remove the unknown numerical prefactor relating Ncorr,T and Tχ. The gray dashed line represents the RFOT prediction [see Eq. (13)]. Both panels (a) and (b) show that the relative evolution of Ncorr in τα is the same for all values of N varying from 1 to 20 and is very similar to the RFOT prediction, on a wide four-decade range of τα.</figcaption></figure>

we draw X(3)(u0) from literature—see Refs. [33,38]—and we select for each N-PG a reference temperature Tref for which fα ≈ 10 Hz (see Table II). Figure 6(a) shows the evolution of Ncorr(T)/Ncorr(Tref) as a function of the relaxation time τα(T)/τα(Tref).

Figure 6 shows that the increase of Ncorr(T)/Ncorr(Tref) upon cooling is identical, within the error bars, for all the oligomers measured, with N rising from 1 to 20. As we shall discuss in detail in the next section, this was not obvious a priori and represents the key result of this work.

To reinforce this result, let us consider another estimation of Ncorr drawn from the linear spectra by derivation of τα vs T. As it was shown in Refs. [60–62], this estimator Ncorr,T(T) is given by

$$N_{\text{corr,T}}(T) \propto T \chi_{\text{T}}(T) = \left| \frac{\partial \ln(\tau_{\alpha}/\tau_0)}{\partial \ln(T)} \right| = T \frac{D T_0}{(T - T_0)^2}, \quad (9)$$
where the proportionality factor relating $N_{corr,T}(T)$ and $T\chi_T$ does not depend on $T$, even if its precise value is unknown. Once again, this can be circumvented by considering the ratio $\frac{N_{corr,T}(T)}{N_{corr,T}(T_{ref})}$, which is plotted in Fig. 6(b) and reads

$$ \frac{N_{corr,T}(T)}{N_{corr,T}(T_{ref})} = \frac{T\chi_T(T)}{T_{ref}\chi_T(T_{ref})}. \qquad (10) $$

We note that the six liquids studied in the present work are in agreement with the previous works of the Augsburg team— see Refs. [15,54,55] reporting the behavior of $|X_3^{(3)}(\omega\tau_\alpha)|$ in temperature of five different glassformers. In these references, it was shown that the maximum over frequency of the dimensionless cubic susceptibility $|X_3^{(3)}(\omega\tau_\alpha)|$ scales in temperature as $E_{act}(T) \equiv |\partial \ln(\tau_\alpha(T))/\partial(1/T)|$. Because one has $E_{act}(T) = T \times |T\chi_T|$, and because the cubic response experiments can be followed on a quite narrow interval in temperature $[T_{min}; T_{max}]$ with $T_{max} \approx 1.15 \times T_{min}$, the difference between our Fig. 6(a) and the results of Refs. [15,54,55] is below the experimental resolution.

Summarizing, Figs. 6(a) and 6(b) show very similar results. We have two independent estimators of $N_{corr}$, the first one obtained from the $3\omega$ measurements, the second one from the linear measurements. Both give similar results: The evolution of $N_{corr}$ as a function of $\tau_\alpha$ is independent of $N$. This is a remarkable result, since the two quantities plotted in Figs. 6(a) and 6(b) come from independent lines of thought. We shall now discuss what this implies for our understanding of the glass transition.

## IV. DISCUSSION

In Fig. 6, although $N$ varies considerably—from 1 to 20—the behavior remains quantitatively the same. As explained above, because the size of our longest oligomer (presumably) exceeds that of typical dynamic heterogeneities, this result was far from obvious a priori, and it establishes that the relative evolution of $N_{corr}(\tau_\alpha)$ is insensitive to the backbone linking the monomers. Besides, as we show now, many of the quantities often used to characterize glassformers around $T_g$ vary in our $N$ range. We shall enumerate them to get a deeper understanding of the meaning of Fig. 6. Finally, we shall inspect what are the predictions of the first-principles theories of the glass transition that we can compare to Fig. 6(a).

### A. Microscopic parameters having no impact on Fig. 6

#### 1. Fragility, glass transition temperature, and spectral width

Figure 7 presents the evolution in $N$ of two parameters widely used to classify glassformers, namely, the glass transition temperature $T_g$ (blue curve, blue dots, left vertical scale) and fragility $m$ (red curve, red triangles, right vertical scale). Both show a very rapid growth between $N = 1$ and 2, and a slower increase between $N = 2$ and $\approx 6$, before reaching a plateau. We have a similar behavior for a third widely used parameter, namely, the stretching, given by $\chi_1$'s FWHM (Fig. 5): We can see a jump between $N = 1$ and 2, and a constant value for larger $N$. Note that the value of $T_g$ for 2-PG, situated higher than the curve, has already been reported in Ref. [32]. Overall, despite the noticeable variations of all these

<figure><img src="image_10.png" /><figcaption>FIG. 7. Glass transition temperatures $T_g$ (blue curve, blue dots, and left vertical scale) and fragility $m$ (red curve, red triangles, and right vertical scale) vs $N$. The error bar on $T_g$ is on the order of $\pm 0.5$ K, yielding the relative error on $m$ to be close to $\pm 3\%$.</figcaption></figure>

three parameters within our range in $N$, the content of Fig. 6 remains unchanged.

#### 2. $\beta$ and other relaxation modes

As detailed in Refs. [32,49], in addition to the $\alpha$-relaxation mode, there are other modes in $N$-PG, in particular the $\beta$ mode, which varies significantly in our range of $N$. We note that this evolution of the $\beta$-relaxation mode does not affect the behavior in Fig. 6. This can be understood because this $\beta$-relaxation mode appears at frequencies much higher than $f_\alpha$, and has a small amplitude. As our frequency experimental range does not extend close to $f_\beta$, we shall not enter in the interpretation of nonlinear susceptibility around $f_\beta$, which has already been reviewed in Refs. [33,54].

#### 3. Densities of OH bonds and strength of dipole-dipole interactions

As shown in Fig. 8, in our range of $N$, the numerical density of OH groups varies by a factor $\approx 15$. This implies that the OH bonds, which are numerous for $N = 1$, are no longer at play for $N = 20$. Similarly, the dielectric strength $\Delta\chi_1$ varies by a factor 12–14 for our $N$-PG series, which implies a drastic change for the typical dipole-dipole interaction $I = \mu^2/(4\pi\epsilon_0a^3)$ for neighboring dipoles $\mu$ separated by a distance $a$. It is of interest to compare $I$ to the mean thermal energy per degree of freedom $k_B T_g$ by defining the ratio $R = I/(k_B T_g)$. Remembering that the dielectric strength is given by $\Delta\chi_1 = \mu^2 g_K/(3\epsilon_0 k_B T a^3)$, we get $R \approx \Delta\chi_1(T_g)/(4g_K)$, which allows to assess the dependence of $R$ in $N$. Indeed, with $g_K(N = 1) \approx 2.85$—see Ref. [53]—we get $R(N = 1) \approx 5$, while $R(N = 20) \approx 1$—for the latter case we took $g_K(N = 20) \approx 1$ because $\Delta\chi_1(N = 20)$ is very small with respect to $\Delta\chi_1(N = 1)$. Finally, as for the OH bondings, we get that, in our $N$ range, the typical dipole-dipole interaction, scaled by $k_B T_g$, disappears for the game for $N = 20$, while it is non-negligible a priori for $N = 1$. Yet, neither of these two qualitative changes has any consequence in Fig. 6.
<figure><img src="image_11.png" /><figcaption>FIG. 8. Number of molecules, monomers and -OH groups per cm³ obtained from the densities given by the suppliers, varying from ≈1.036 to 1.00 g/cm³ for N = 1–20. Apart from the simple behavior (constant or ∝1/N), there is a slight bending for the range 1 ≤ N ≤ 3, which could contribute to the variations of T_g and m for N ≤ 3.</figcaption></figure>

## B. Prediction of the various first-principles theories of the glass transition for Fig. 6

We now move to a comparison with theoretical predictions. We limit ourselves to discuss theoretical frameworks constructed to account for the glass transition in itself. We leave aside the various phenomenological models taking the viscous slowdown for granted and attempting to model the cubic nonlinear responses. As shown in Ref. [13], most of these models (if not all) implicitly require the growth of dynamically correlated domains upon cooling, and are consistent with the general prediction of Bouchaud and Biroli expressed in Eq. (2).

### 1. Random first-order transition theory

RFOT is a first-principles theory of glass formation inspired by p-spin models [63]. Its main output is that the physics of supercooled liquids is driven by the emergence of amorphous order, i.e., by nonperiodic configurations of very low free energy [18,64–67]. The typical size $\xi_{stat}$ of these amorphously ordered domains—called glassites—increases upon cooling, because of the decrease of the configurational entropy $s_c(T)$ with T. As a result, upon cooling, the free energy term $\propto s_c(T)\xi_{stat}^3$, pushing any glassite to “melt,” i.e., to change its amorphously ordered configuration, needs larger values of $\xi_{stat}$ to counterbalance the “pinning” free energy term $\propto \xi_{stat}^{3/2}$ which, because of boundary conditions, pushes the glassite to remain in its amorphously ordered initial configuration [18,64–67].

In RFOT, $\xi_{stat}(T)$ is a priori different from the dynamical correlation length $\xi_{dyn}(T)$ obtained from the four points correlation function defining the size of DHs, and on general grounds $\xi_{stat} \le \xi_{dyn}$. In practice, as the dynamics is driven by the glassites themselves [64], $\xi_{stat}/\xi_{dyn}$ is a constant of order one close to $T_g$, and RFOT states [18,66,67] that the number

<table><caption>TABLE III. Calculation of the crossover value $N^*$ within RFOT: The $\Delta C_p$ values are from Ref. [68] and must be converted in J/(mole.K) before using Eq. (12) to get the number of beads per oligomer. As the density of N-PG is—mainly—independent of N, the decrease of $\Delta C_p$ when N increases explains why the number of beads per oligomer increases much less than linearly in N—fourth column. This finally leads to a number of monomer per DH increasing with N—last columns.</caption><thead><tr><th rowspan="2">N</th><th rowspan="2">ΔC<sub>p</sub><br>J/(g K)</th><th rowspan="2">m</th><th>n<sub>beads</sub><br>per<br>oligomer</th><th>n<sub>oligomer</sub><br>per<br>DH</th><th>N<sub>mono</sub><br>per<br>DH</th><th colspan="2">N* from Eq. (3)<br>with prefactor</th></tr><tr><th></th><th>= 0.9</th><th>or = 1.8</th></tr></thead><tbody><tr><td>1</td><td>0.87</td><td>51</td><td>4.4</td><td>43.9</td><td>43.9</td><td>4.9</td><td>or 9.8</td></tr><tr><td>3</td><td>1.07</td><td>77</td><td>9.0</td><td>21.6</td><td>65</td><td>5.9</td><td>or 11.8</td></tr><tr><td>7</td><td>0.70</td><td>82</td><td>12.4</td><td>15.7</td><td>110</td><td>7.5</td><td>or 15</td></tr><tr><td>17</td><td>0.64</td><td>84</td><td>26.2</td><td>7.45</td><td>127</td><td>8.0</td><td>or 16.0</td></tr><tr><td>34</td><td>0.600</td><td>84</td><td>48.7</td><td>4.00</td><td>136</td><td>8.2</td><td>or 16.4</td></tr><tr><td>68</td><td>0.556</td><td>84</td><td>90.0</td><td>2.17</td><td>147</td><td>8.5</td><td>or 17.0</td></tr></tbody></table>

$N_{corr}$ of correlated beads is given by

$$N_{corr}(T) = \left[ \frac{8}{3\sqrt{3\pi}} \frac{\ln(\tau_{\alpha}(T)/\tau_0)}{\ln(\alpha_L a^2/(\pi e))} \right]^2, \quad (11)$$

with $\tau_0$ the microscopic time defined above in Eq. (4); a the size of a bead, i.e., the size of an effective degree of freedom deduced from the molecular latent heat of crystallization; and $\alpha_L \simeq (10/a)^2$ the effective Einstein oscillator force constant acting on a bead. Reference [69] gives the definition of beads, as well as the number of beads per oligomer $n_{beads}$, provided that the fragility $m$ and the specific heat jump per mole $\Delta C_p$ are known:

$$n_{\text{beads}} = \frac{34.7 \times \Delta C_p}{m \times S_{\text{LJ}}} \quad \text{with } S_{\text{LJ}} = 1.68 \text{ kB} \simeq 10 \text{ J/mole.} \qquad (12)$$

Because in Eq. (11), the dependence on microscopic parameters is only logarithmic, and because $\tau_\alpha(T_g)/\tau_0$ is so large, one of the RFOT’s prediction [18,66,67] is that the number of correlated beads is a quasiuniversal number $\simeq 195$ at $T_g$. Table III shows that, by using the data measured by Johari *et al.* [68] we can draw from Eq. (12) the number of beads per oligomer, given in the fourth column of Table III. By using the RFOT prediction that 195 beads are dynamically correlated at $T_g$, one finds the number of oligomers contained in any typical DH at $T_g$ (see the fifth column of Table III). By multiplying this result by N, we find the number “$N_{mono}$ per DH” of monomers contained in a DH at $T_g$ (see the sixth column of Table III). Finally, this result can be inserted in the right-hand side of Eq. (3) to get the sought value of $N^*$, which value depends of the chosen prefactor in Eq. (3), i.e., on whether one compares the DH diameter to $R_{ee}$ of to $D_{sph}$. A self-consistent analysis of the last column of Table III shows that $N^* \simeq 16$ in the former case, while $N^* \simeq 7$ in the latter case. We shall discuss briefly these values in the conclusion.

We now use Eq. (11) to get the RFOT prediction relevant for our experimental Fig. 6. We get

$$\left[ \frac{N_{\text{corr}}(T)}{N_{\text{corr}}(T_{\text{ref}})} \right]_{\text{RFOT}} = \left[ 1 + \frac{\ln(\tau_{\alpha}(T)/\tau_{\alpha}(T_{\text{ref}}))}{\ln(\tau_{\alpha}(T_{\text{ref}})/\tau_0)} \right]^2, \quad (13)$$
which implies that the experimental quantities plotted in Fig. 6 (1) should fall down onto a master curve—owing to the logarithmic dependence in $\tau_0$ and (2) that this master curve should be indistinguishable from a straight line if the experimental resolution is $\pm 5\%$ as in our Fig. 6(a)—owing to the fact that the ratio of logs is smaller than 0.16 in absolute value. Both features are observed in Fig. 6, where the RFOT prediction is plotted as a dashed black line, showing that RFOT is quantitatively consistent with our experimental behavior.

## 2. Frustration theory

Frustration theory is another first-principle framework [70] making the assumption that a locally preferred structure (LPS), maximizing the local density, plays the key role around the crystallization temperature $T_m$. Frustration comes in because this LPS cannot tile the three-dimensional space [70]. Upon cooling, the viscous slowing down arises because of the growth of the size $\xi_{LPS}$ of LPS domains surrounded by defects that cost energy. It turns out that general predictions can be done [70] by mapping the problem to the abstract space $\mathcal{E}$ which metrics is curved just enough to allow the LPS crystallizing at a temperature $T^*$. One distinctive general prediction is that, even though $\xi_{LPS}$ grows upon cooling below $T_m \le T^*$, there is a low temperature $T_{sat}$ below which $\xi_{LPS}$ saturates, because frustration is so strong that there is no way to further optimize the free energy. As a result, $\tau_\alpha$ is predicted to recover a simply activated behavior below $T_{sat}$.

Frustration theory is thus really different from RFOT, where one predicts that $\xi_{stat}$ diverges at the Kauzmann temperature $T_K < T_g$. However, even if some detailed calculations exist in model systems [71,72], predictions in realistic cases are limited in Frustration theory, mainly because the nature of the LPS is unknown and may depend on molecular details—the only known case is that of liquids of spheres, for which the LPS is an icosahedron [70]. As $\tau_\alpha(T)$ remains superactivated at least down to $T_g$, $T_{sat}$ must be below $T_g$, i.e., nonlinear experiments below $T_{sat}$ would be extremely long and difficult. Moreover, as LPS—assuming that they exist—are extremely complicated molecular arrangements, they may well couple to the applied field in a way very similar to that of amorphously ordered glassites put forward by RFOT. Indeed, we shall advocate in the next section that the crucial point to account for nonlinear experiments is that collective effects play the central role, which, as we shall argue, is much more a demanding requirement than mere dynamical correlations. These collective effects amount to get some molecular configurations which, because they have a low value of free energy, react collectively to the applied field, because of the rigidity that is the counterpart of their collective optimization.

To conclude our comparison of Frustration theory predictions with our Fig. 6, it is possible that this theory is consistent with our data, but we cannot push our experiments in the range where the distinctive features of Frustration with respect to RFOT can be inspected, consistently to what some of us had already discussed in Ref. [38].

## 3. Kinetically constrained models

Kinetically constrained models (KCM) are a family of models [73–76] assuming that the thermodynamics of super-

cooled liquids is trivial, i.e., that the Hamiltonian $\mathcal{H}$ boils down to that of an ideal gas of particles, and that the nontrivial dynamical rules only matter to account for all the glass phenomenology around $T_g$. At first sight, this may seem strange, since one may ask how nontrivial dynamical rules may arise when the hamiltonian is trivial? This is nonetheless possible, as it has been analytically demonstrated on the so-called plaquettes models [77,78]: To give a simple illustration, consider a two-dimensional space tiled with, e.g., triangular plaquettes [77], each containing three physical spins $S_{i1}$, $S_{i2}$, and $S_{i3}$ at its vertex. Then, assume that each physical spin interacts exclusively with the two other spins of the same plaquette $i$. This Hamiltonian $\mathcal{H}(\{S_{ik} = \pm 1\})$ is complicated, since each spin belongs to the two plaquettes that it connects. However, it was demonstrated that this Hamiltonian can be mapped to another one involving only the plaquettes variables $\sigma_i = S_{i1}S_{i2}S_{i3}$, and that $\mathcal{H}(\{\sigma_i\}) = \sum_i \sigma_i$, i.e., we have a trivial “ideal gaz” of plaquettes variables. As for the dynamical rules [77] of the plaquettes variables $\sigma_i$, they are the result of the thermal fluctuations of the physical spins $S_{ik}$. It turns out that these rules are complex, since the ability of $\sigma_i$ to relax to its lowest energy state depends on the state of its neighboring plaquette variables. This flip may happen only in the case where at least one of the neighboring plaquette variable is in its excited state. This is a quantitative illustration of the key intuition of the KCM models [74]: As the excited states $\sigma_i = 1$ are rare at low $T$, they can be thought as “empty sites,” i.e., sites where the free volume lies. As a result, it is physically very appealing that relaxation around $T_g$ only happens in the sites having at least one empty neighbor (i.e., a neighbor where $\sigma = 1$). Besides, as the average concentration $c_{exc}$ of empty sites is very small at low $T$, this creates long relaxation time [79,80] as regions containing as much as $1/c_{exc}$ sites have to “share” the same excitation; i.e., the full relaxation of the region will be over only when the same elementary piece of free volume will have come around each site. Finally, as $c_{exc} \ll 1$, one naturally expects [74,80] strong spatial heterogeneities of the dynamics: The faster regions are those which, at a given time, are richer than the average in empty sites. To summarize, KCM models naturally account [73,76] for the main features of glassy dynamics around $T_g$.

Do KCM models account for the nonlinear response behavior such as those shown in Figs. 4 and 6(a)? This question is still debated and opposite views coexist in the literature [38,52,81–83]. The difficulty lies in the fact that there is no analytical way to derive explicitly cubic responses $\chi_3(\omega, T)$ starting only from an expression of $\mathcal{H}$ as given by KCM models. Some works claim [82,83] that, starting from the main KCM ideas, one can cook equations accounting—semiquantitatively—for the experimental behavior of cubic responses. On the other hand, some general arguments have concluded in the opposite way [38,52]. Importantly, a recent numerical work [81], in which one of us is involved, simulates explicitly a very simple KCM model and finds that its cubic responses $\chi_{3,KCM}(\omega, T)$ do not show the main experimental features: $|\chi_{3,KCM}(\omega, T)|$ has no peak at finite frequency and decreases monotonously when increasing the frequency; besides, the dimensionless cubic susceptibility $|\chi_{3,KCM}(\omega, T)|$ is strictly independent of $T$ when plotted as a function of $\omega\tau_\alpha$. This is at odds of our Figs. 4 and 6(a).
How is it possible that a KCM model, as that of Ref. [81], is
able at the same time to account for dynamical heterogeneities
and not for the nonlinear behavior? As explained in Ref. [81],
this comes from the difference between dynamical correlation
effects and genuine collective effects. Collective effects arise
when it exists some molecular configurations $C_{\text{opt}}$ of low free
energy $\mathcal{F}_{\text{opt}}$, i.e., preferred configurations where the interac-
tions between $\approx N_{\text{corr}}$ molecules lead to a significant lowering
of $\mathcal{F}$. Even if such preferred configurations are aperiodic, even
if they do not yield the absolute minimum of $\mathcal{F}$, even if they
only involve a finite value of $N_{\text{corr}}$, there is, nonetheless, a
trend to come back to such preferred configurations when the
system is submitted to an external field. Because a configura-
tion $C_{\text{opt}}$ arises by efficiently combining interactions between
$\approx N_{\text{corr}}$ molecules, the coupling to an external field will be
larger for larger $N_{\text{corr}}$, yielding finally an overall behavior of
$|X_3^{(3)}|$ tracking that of $N_{\text{corr}}$. In a KCM model, however, one
starts by assuming that the value of $\mathcal{F}$ negligibly depends
on the considered configuration [73,76]: Therefore, there is
no trend to come back to any configuration, and, within a
given DH of $N_{\text{corr}}$ molecules, the field couples only to one
molecule at a time, namely, the one having the opportunity
to relax because it has an empty neighbor. Hence, the overall
behavior where $|X_3^{(3)}(\omega\tau_\alpha)|$ is a constant in temperature, as in
any noninteracting model.

Therefore, despite its extremely simplified character, the
KCM model of Ref. [81] has the great interest of giving an
example where Fig. 6(a) cannot be accounted for—as just
explained—while Fig. 6(b) can be accounted for. Indeed,
because it involves only ln(τ<sub>α</sub>(T)) and its derivative in tem-
perature, Fig. 6(b) is recovered by any model accounting
well enough for the behavior of ln(τ<sub>α</sub>(T)). This illustrates
that χ<sub>3</sub>(ω, T) and N<sub>corr,T</sub> come from distinct lines of thought,
as already emphasized (see Refs. [30,62]). The fact that the
overall behavior of Figs. 6(a) and 6(b) is very similar is thus
an extremely nontrivial result and a strong argument in favor
of thermodynamic theories stating that order effects—either
amorphous or LPS driven—play a key role in the glass transi-
tion.

V. CONCLUSION

We performed linear and 3ω dielectric measurement on a
family of oligomers, N-PG, with N ranging from 1 to 20, in
order to investigate the influence of the length of the chain on
the glass transition properties. The parameters obtained from
our linear measurements are in good agreement with those
reported in previous results [31,32,48,49]. The influence of
N on the basic properties such as T_g and m is clear for N ≤ 3,
but is marginal as soon as N > 3.

We showed that the dielectric linear susceptibility $\Delta\chi_1$
varies as 1/N. As the possible variation of the Kirkwood
factor with N is far from explaining this effect, the simplest
interpretation would be to imagine that the only contribution
to the polarization would be due to the two ending OH groups,

no other contribution coming, in our measurement conditions,
from the chemical units situated in-between. In this scenario,
varying the length of the molecule from 1 to 20, we do not
perturb the dipoles. This is an unexpected and nonobvious
behavior, which has not yet received its microscopic justifi-
cation.

With a molecule length extending from 1 to 20, our MD
simulations allow to know that we explore either side of the
DH mean diameter $D_{\text{corr}}$ close to $T_g$. We naturally expected
a modification of the evolution of characteristic parameters
around $D_{\text{corr}}$, like the fragility $m$, $T_g$ or even the relative evolu-
tion of $N_{\text{corr}}$ in temperature. To locate precisely this crossover,
the difficulty lies in predicting the absolute value of $D_{\text{corr}} \propto
N_{\text{corr}}^{1/3}$ for which very different experimental estimates coexist
in the literature. We note however that within RFOT, we come
to a prediction that the crossover is expected for N-PG chains
of length $N^*(\tau_\alpha = 10 \text{ ms}) \simeq 7$, which amounts, by the same
reasoning, to $N^*(\tau_\alpha = 100 \text{ s}) \simeq 8.4$. This is nearly three times
larger than the characteristic chain length $\approx 3$ beyond which
$T_g(N)$ and $m(N)$ no longer evolve, as observed in Fig. 7. But
it is difficult to disentangle the various effects, as $T_g$ and $m$ are
also correlated to the volume density of monomers.

In contrast, the absence of influence of the chain length in
Fig. 6 is striking. Using two independent estimators, which
reinforce each other, we showed that the relative evolution
of $N_{\text{corr}}$ versus $T$ is independent of $N$. Consequently, the
results presented in Fig. 6 depend neither on $T_g$, nor on $m$,
nor on the spectral width, nor on the $\beta$-relaxation intensity,
nor on the volumic density of H bonds, nor of the strength of
dipole-dipole interactions, nor on the length of the chain, even
when we approach the crossover value $N^*$ introduced in this
work, where one may have expected an effect of the molecular
skeleton. Thus, this plot unveils a very robust property close
to the glass transition. Besides, as discussed above, Fig. 6(a)
is of great physical interest since it cannot be accounted for by
some of the first-principles theories of the glass transition, at
variance with Fig. 6(b).

ACKNOWLEDGMENTS

We warmly thank Nicolas Delpouve for long-term collab-
oration and Marceau Hénot for his introduction to numerical
methods in open-source Molecular Dynamics software. Be-
sides, we acknowledge financial support by IRAMIS, as
well as by the French Government under the France 2030
program, reference ANR-11-IDEX-003 (Interglass Project of
the PHOM Department of University Paris-Saclay), and by
Agence Nationale de la Recherche, reference ANR-24-CE30-
0442 (GLASSGO project).

DATA AVAILABILITY

The data that support the findings of this article are not
publicly available. The data are available from the authors
upon reasonable request.
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