Understanding the role of dendrimers in cancer treatment and diagnosis requires knowledge of their basic properties and applications. Their unique architecture and properties place them in a different class of polymers—dendritic polymers [3]. Dendrimers are highly branched, open covalent assemblies of branch cells (monomers) radially attached to a core in successive layers or generations [4]. Figure 1 below depicts a typical schematic of the structure.

Their architecture rests on three main components: an initiator core, the branch cell amplification region, and finally the outer terminal functional shell [3]. The core is often described as the information center of the molecule and ultimately defines the characteristics of the dendrimer. The amplification region determines the features of the bonds between the dendrimer and guest molecules through the void density and composition of the branch cells. Finally, the exterior layer represents the outer surface of the last shell and is composed of terminal functional groups used for various purposes, such as the foundation for subsequent shells or generations and the regulation of the flow of guest molecules into and out of the dendrimer [5]. It is worth noting that a dendrimer is not limited to spherical shapes but may also be found as a disk.

More insight on the unique nature of dendrimers can be gained by exploring how they differ from other “conventional” synthetic polymers categorized as linear, cross-linked, and branched. While all of these types of basic monomeric blocks instill a large variety of properties, the arrangement of the blocks differs greatly. Dendrimers show very little branching or crosslinking and are almost entirely characterized by their hyperbranched structure [5]. These architectural differences have significant effects on their physical and chemical properties even when these polymers are composed of comparable monomer building blocks. In contrast to linear polymers, dendrimers show predictable shape variations instead of random coil configurations, occupy less hydrodynamic volumes, and have very low polydispersity. Dendrimers are also non-crystalline, amorphous substances exhibiting no repetition and a Newton type rheology with a lower viscosity that reaches a plateau at the fourth generation before declining [6]. Higher solubility is achieved along with isotropic electronic conductivity. More subtle differences can be observed when considering other dendritic structures. For instance, a constant density changes as a function of generation number for dendrimers with asymmetric branch cell arrangements; whereas, symmetrical branch cell arrangements achieve a minimum density between generations 4 and 7. Dendrimers are also unique among other polymers, because they experience exponential growth of the terminal functional group and maintain consistent shape as a function of generation number [5]. Finally, dendrimers can be considered the link in properties and architecture between thermoset and thermoplastic polymers [5].

The functionalities and characteristics of dendrimers logically lead to a comparison with biological polymers or proteins. While only a few examples of dendrimer architectures are found in nature where the principal equivalent would be glycogen (involved in the storage of energy) and proteoglycans (involved in the absorption of energy), several similarities can be noted [5]. Both types are globular (when comparing to globular protein types), comparable in sizes, and composed of defined monomeric units arranged in a distinct architecture. Moreover, as is often the case with globular proteins, precise void space created by the folding of peptic chains afford these molecules special capabilities that can be imitated by the dendrimers’ ability to host microenvironments within branch cells with properly designed active sites [7]. Furthermore, through non-covalent forces, dendrimers can exhibit a “lock and key” chemical behavior familiar to many proteins. Although many parallels were drawn between proteins and dendrimers a few differences should be mentioned. Dendrimers are more customizable through their synthesis and can be designed to achieve a wide variety of goals. Indeed, because of their architecture (radially attached branch cells), a greater surface area is available for the terminal functional groups. These terminal branches become less flexible with an increase in generation number due to crowding of the outer layer [5].

In 1985 Tomalia et al. [8] first discovered PAMAM dendrimers. They are the first family of dendrimers that have been synthesized and commercially available. PAMAM dendrimers are synthesized starting from an alkyldiamine core reagent. By reacting with the Michael addition of methylate, the smallest PAMAM dendrimers could be obtained [8]. Higher generations of PAMAM could be synthesized by a reaction of the small dendrimers with a large excess of ethylenediamine. Starting from G7, the de Gennes dense packing effect constrains the growth of dendrimers [9]. The growth of PAMAM dendrimers with generations higher than G10 is inhibited. Various generations of PAMAM dendrimers have specific size and shape characteristics. Their molecular weights are normally narrowly distributed with a high molecular uniformity. Their surface functional groups are ideal for potential chemical and biological reactions and applications [10].

The structural properties of dendrimers to be explored by SANS include: inter-molecular structure, intra-molecular cavity, radius-of-gyration (R_G), effective charge number of a single dendrimer molecule, and water penetration into the interior of the dendrimers.

The structural information of SANS experiments is obtained by the comparisons between the experimentally detected spectra and the theoretical results of the system. The theoretical results were calculated with a reasonable effective potential V_eff(r) developed based upon statistical mechanics models such as the Ornstein-Zernike (OZ) integral equation theory. Neutrons scatter heavily (stronger signals from the experimental result) with the proton nucleus in the experimental system. They also scatter almost evenly with the other nucleus in the system. Neutron scattering has the unique capability of determining structural information of certain parts of the system through labeling (e.g., deuterating and/or protonating some parts of the dendrimers).

It is important to realize that neutron scattering experiments prefer the use of the solutions of D₂O over H₂O. This is due to the fact that hydrogen has such a large cross section, and the scattering signals from the hydrogen in H₂O molecules will cover up most contributions from the macromolecules. However, for a macromolecular solution, while substituting the D₂O solvent for the H₂O solvent at a condition [H ⁺] = [D ⁺], the pH of a H₂O solution is different from the corresponding pD in a D₂O solution. It is generally agreed that pD = pH + 0.41 [11]. Furthermore, the concentration of deuteron in solution is corrected by the measured deuteron activity [12]. The concentration of added acid used to protonate the amino groups, such as deuterium chloride, is represented by the acidity scale factor α , which is defined as the molar ratio of acid to primary amine terminal groups:

Where n is the dendrimer generation. The word “protonation” is used to describe the binding between the deuterons and the amino groups. For typical experiments, the dendrimers were dissolved in D₂O solvent at different levels of molecular protonation.

For the data analysis, models for the SANS absolute intensity I(Q) have been developed [13,14] for dendrimer study, where the scattering intensity, I(Q) is a function of the momentum transfer in reciprocal space, the scattering vector . Q is a scalar connected to the scattering angle, θ, and the wavelength of the radiation, λ. Q is at the range of 10⁻³–1 Å⁻¹ in a typical SANS experiment with a selected sample-to-detector distance. The size of the region investigated in real space is related to Q by , and falls in the range of 3–12 Å [15]. I(Q) depends on the atomic features and the position of the atomic scattering centers [16]. Normally the predicted theoretical intensity distribution is utilized to fit with the SANS measured coherent scattering intensity . Results from the fitting process can be used to compute key parameters related with the structure of macromolecules, such as radius of gyration.

The SANS measured coherent scattering intensity is corrected for detector background and sensitivity and for scattering contributed from empty cells. It is placed on an absolute scale according to a standard procedure described in [13,14,17,18]. Thereafter, is modeled by the following integral equation:

where is the theoretical intensity distribution, δ(Q) is the width of the experimental resolution function at Q, and Q_m is the mean Q value. The structural information of dendrimers could be obtained by the comparisons between the experimental spectra of and the theoretical results . The theoretical results were calculated with the commonly used monodispersed centrosymmetric core-shell model with diffusive interfaces [19] developed with statistical mechanics models such as the Ornstein-Zernike (OZ) integral equation theory. The integrity of the structural information provided by these theories was validated by MC or MD computational simulations [20]. can be further calculated by the so-called factorization approximation [17,21]:

where A is the scattering contrast between dendrimer particles and the solvent depending on the contrast, particle number density, and the particle volume, P(Q), the normalized intra-dendrimer structure factor, and S(Q), the inter-dendrimer structure factor. By assuming a diffuse interface between the solvent and dendrimer, P(Q) is given by the modified fuzzy ball model which approximates the PAMAM molecular density profile by the convolution of a hard sphere with radius R and a Gaussian with variance σ² characterizing the soft shell region [13,22,23], and S(Q) is obtained by numerically solving the OZ integral equation for one component system in combination with the hypernetted chain (HNC) closure [24]. The effective inter-dendrimer interaction is approximated by the screened Coulombic potential and the essential physical quantities characterizing the charged PAMAM solutions, such as the effective charge carried by a single PAMAM dendrimer and the ionic strength, which can be converted from the fitted potential parameters based on the generalized one-component macroion approach (GOCM) [25]. In this approach, the radius of gyration R_G can be calculated and expressed as a simple analytical function of R and σ:

The normalized radial intra-molecular density distribution function, ρ(r), is directly calculated by R and σ during calculation of P(Q). The pair correlation function, g(r), which gives the total correlation function between two given dendrimer molecules, may be calculated by S(Q) through definition: (FT: Fourier transform). Moreover, S(Q) can lead to effective charge number of a single dendrimer molecule, C_eff [13]. Thus, fitting with theory calculated will allow us to at least extract the following parameters: P(Q), S(Q), R_G, ρ(r), g(r), and C_eff.

Figure 2 presented Liu et al.’s work [26] for this data analysis methodology, which is an experiment for PAMAM dendrimers at three different levels of pH values (molecular protonation levels α = 0 (Figure 2(a)), 1 (Figure 2(b)), and 2 (Figure 2(c))) and two different acids (blue symbols: deuterated sulfuric acid, and red symbols: deuterium chloride). It shows the good agreement between SANS intensities and model fitting results.

By using the SANS data analysis methodology, Chen et al. [13] studied the SANS results of charged G4 dendrimer solutions in various thermodynamic states by inter-dendrimer structure factor S(Q) and the intradendrimer structure factor P(Q). They proposed an instructive explanation about the counterions’ effect on the conformation and structure of dendrimers and illustrated the effect of pD on the conformation of PAMAM dendrimers. They found that the number of counterions connecting with dendrimers increased first with the addition of DCl, while the effective charge remained invariant with a further increase of molecular protonation. Based on SANS data analysis, they explored that the swelling of the molecule occurs upon the accretion of the acid, notwithstanding the dependence of the conformation on the pD is about 4%. An internal structural change of the dendrimer was found during protonation process. In 2008, Chen et al. studied using SANS the intramolecular cavity of G4-6 PAMAM dendrimer [21]. They found a significant amount of water inside the dendrimers. By changing the hydrogen-deuterium ratio of the solvent, obtaining the scattering contribution from the water and dendrimer components and combining it with an analytical model, the intramolecular space of PAMAM dendrimer was quantified. The internal cavity increases with the increase in the generation of dendrimers, while the water volume of a dendrimer remained invariant. In addition, they found the water density in dendrimers is higher than bulk water. Still in 2008, Chen et al. investigated the structure changes of the PAMAMs of different generations [23]. They found that with an obvious change of the intramolecular density profile, from neutral pD to acidic environment, the dendrimers’ structure changes from a diffuse structure to a hard sphere, which means the molecular size remains invariant with a pD change. This finding suggests the PAMAM’s molecular sizes have a weak dependence on the protonation effect. They also discovered a close relationship between generation and the degree-of-counterion-condensation. The chloride condensation is closely related to the higher generation of PAMAMs. In 2009, another paper by Chen et al. about the effect of counterion on the intermolecular structure and the properties of the conformation reported a surprising transformation of intra-molecular configuration with the use of DBr [27]. From fitting their data to the SANS model, they found out that the charged G5 PAMAM, which is protonated by DBr, carried a less effective charge than that of DCl when α > 1. This work reveals that the specific counterion association is greater for the DBr G5 PAMAM than DCl G5 PAMAM. In other words, under the same level of molecular protonation, bromides have a much greater tendency to affiliate with the charged G5 PAMAM than chlorides. A similar conclusion was reached in another work from 2008, in which it was determined that the molecular size remains invariant with a pD change protonated by either DBr or DCl. Based on previous work, Chen et al. [23] continued studying the structural characteristics of PAMAMs by SANS. One year later, they illustrated the counterion valence effect on the pH responsiveness and its impact on the structure within a G5 PAMAM [28]. They discovered that the effective charge of PAMAMs is greatly reduced when they are associated with a large amount of divalent counterions; however, no obvious change of the intramolecular density was observed when the dendrimers were charged by D₂SO₄, conflicting with Chen’s previous results for DCl charged dendrimers. The size of dendrimer molecules was validated, once again, as invariant with an increase in protonation. For G4 PAMAM, Chen et al. carried out a SANS experiment in 2010 [22]. They first provided the experimental results to verify an intramolecular structure change in the dendrimers core in order to show that the radial density profile decreases upon an increase in protonation, while the overall size remain the same. Using SANS in combination with SAXS, Chen et al. also investigated the structure characteristics of high-generation (G7 and G8) PAMAMs [29]. They reported an obvious molecular size increase along a continuous variation of the density profile of intramolecular structures with an increase in protonation. This work proposed a new picture of the structural characteristics of high generation PAMAMs and strongly supported the segmental back folding mechanism. This work opposed the current beliefs, which suggested the high generation PAMAMs are supposed to be crowded to hinder to local movement and electrostatic swelling. Most recently in 2011, Liu et al. investigated the G5 PAMAM in D₂O with SANS, NMR, and PFG (proton pulsed filed gradient diffusion) [26]. In this work, NMR suggests a significant change of the hydrodynamic radius with increasing of protonation; however, the SANS results indicated little dependence between them, while displaying the transition from a dense core to a uniform hard sphere when the dendrimer is protonated by DCl. The inconsistency in these results repudiates the validity of using the Stokes-Einstein relation to obtain the size of a polyelectrolyte dendrimer from dynamical measurements, although it is fairly accurate in determining the size of hard colloids.

Additionally, with the use of SAXS and SANS, Gröhn et al. studied the formation of gold nanoclusters inside dendrimer templates [30]. Maiti et al. studied the atomistic structure of ethylenediamine cored PAMAM family up through the 11th generation by SAXS and SANS [31]. By studying the G8 PAMAM by SANS, Nisato et al. found the dendrimer of a uniform sphere with a radius of gyration R_G = 4.0 ± 0.15 nm. They discovered the size is not dependent on the charge density or ionic strength of the solvent, which is contrary to the predictions of current models and simulations [32]. Liu et al. investigated the structural characteristics of G7 and G8 PAMAMs as a function of molecular protonation by SANS and SAXS. Their work revealed an increase of the molecular size and a continuous change in the intramolecular density profile [29]. Liu et al. also executed a study of the structural characterization for G5 PAMAMs by analyzing the SANS data [28].

QENS is ideally suited to investigate both the segmental relaxation and local relaxations in polymer blends [15]. In a QENS experiment, the energies of the neutrons before and after the scattering event are differentiated, and the difference between them provides information on the time scale of motion within the sample. By combining the dominant cross-section of the macromolecules with the use of spectrometers having different energy resolutions, the local motion of macromolecules over a wide range of time-scales, from picoseconds to tens of nanoseconds, can be studied thoroughly.

QENS experiments measure the Incoherent Scattering Function (ISFω). For dendrimers in solution, the ISFω can be considered as the convolution of the dendrimer global, center-of-mass translational Brownian motion with its internal local motion [33]. In order to investigate the internal dynamics of a dendrimer, it is necessary to separate the local motion from the global translational diffusion using the following operation:

where is the total incoherent scattering function. is the contribution from internal motion, which will be convoluted with , the global translational diffusion. The Debye-Waller factor, sample scattering strength, and all the other factors corresponding to the normalization of the scattering intensity are collected into the parameter N. B(Q, ω) is the energy-dependent background that has been fitted with a linear function. can be described by a Lorentzian function with half-width at half-maximum (HWHM) of , where the self-diffusion coefficient D_S can be obtained from NMR experiments. is the sum of an elastic contribution and a quasi-elastic Lorentzian contribution with an HWHM of , and A₀(Q) here is the fraction of the elastic component that is Q-dependent..

Finally, the measured intensity , which is a convolution of with the instrument resolution function R(Q,ω), obtained from pure vanadium measurement, is represented as:

Where R(Q, ω) is the resolution function of the instrument.

This data analysis methodology has been developed by Chen et al. and tested for PAMAMs in solutions of DCl and D₂O at three different levels of pH values (molecular protonation levels α = 0, 1, and 2) [34]. Figure 3 presents an example of the good agreement between QENS intensities and model fitting results [34].

By combining QENS and high-resolution solution NMR spectroscopy, Chen et al. [34] first investigated the G5 PAMAMs as a function of protonation from a dynamical perspective and provided the first conclusive evidence from experimentation with D₂O. QENS clearly exhibits the local motion of dendrimer segments, illustrating an increased local motion with an increase in the molecular charge, which is contrary to the hypothesis that increased protonation will stiffen the dendrimer and hinder the local motion.

They used the diffusion coefficient D and the sphere radius R to describe the confined diffusive motion of dendrimers. QENS measurements show invariant R at different protonation levels specified by α or pD values (Figure 4(b)) [34]. However, as seen from Figure 4(a) (the diffusion coefficient at different protonation levels: blue dots as the protonation level of α = 0, red dots as α = 1, and grey dots as α = 2), the diffusion coefficients of α = 2 are higher than that of α = 1, and the diffusion coefficients of α = 1 are higher than that of α = 0. This tells us despite the fluctuations at different Q values, the internal diffusion coefficient D is seen to increase steadily with the increase in dendrimer protonation [34].

This tendency of D demonstrates that the local diffusive motion is faster when the dendrimer is charged. It is believed that the induced protonation will result in the interaction with the counterions and water molecules. Additionally, intra-dendrimer space will be available for electrostatic swelling and mass redistribution inside the dendrimer.

With the exception of SANS, dendrimers may be characterized by various techniques; the most wildly used scattering techniques include but are not limited to small angle X-ray scattering, small angle neutron scattering and light scattering.

Small-angle X-ray scattering (SAXS) and SANS data analysis are similar in many ways even though SANS uses neutrons and SAXS uses X-rays. For SAXS, the scattered X-ray signals by a given sample are recorded at very small angles. The signals within this angular range take the structural information of dendrimers to be explored. Prosa et al. used SAXS to characterize the single-particle scattering factors produced by different kinds of dendrimers [35]. They also examined the progression of intramolecular organizations of PAMAMs by SAXS and found an internal structure evolution from “star” to “sphere” organizations [36]. By using SAXS and neutron spin-echo spectroscopy Rathgeber et al. studied a single dendrimer for the dynamic and static structure factors and proposed a compact globular shape of dendrimers at least down to the G3 [37]. Mallamace et al. studied the internal density profile in a dendrimer system of various generations by SAXS, and proposed an internal self-similar structure for low generation dendrimers along with a spherical homogenous dense structure of higher generation [38].

Dynamic light scattering (DLS) is also widely used for studying the structure of macromolecules [36,38,39,40,41,42]. The measured data in DLS experiment are the correlation curve. Embodied within the correlation curve is all of the information regarding the diffusion of particles within the sample being measured. The diffusion coefficient (D) is calculated by fitting the correlation curve to an exponential function, with D being proportional to the lifetime of the exponential decay. Different mathematical approaches can be employed to determine D, such as: the cumulant method [43] and CONTIN algorithm [44]. D will then be used to calculate the hydrodynamic radius (R_H) of dendrimers through the Stokes–Einstein equation.

where k is the Boltzmann constant, T is the temperature, and η is the medium viscosity. With small differences, the R_H derived by DLS is comparable with R_G in most cases. By using DLS Choi et al. obtained the mean diameter of PAMAM dendrimers to be around 200 nm [45]. Mourey et al. obtained the polyether dendrimers’ molecular weight by Laser light scattering (LALLS), which agreed with the anticipated values determined from the synthetic method [46]. The following equation is used to obtain the average molecular weight from LALLS:

Here R_θ(υ) is excess Rayleigh scattering at retention volume υ, K is an optical constant, and m is the total mass of sample [46].