Helmholtz free energy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Helmholtz_free_energy 

In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which temperature is held constant. At constant temperature, the Helmholtz free energy is minimized at equilibrium.

In contrast, the Gibbs free energy or free enthalpy is most commonly used as a measure of thermodynamic potential (especially in chemistry) when it is convenient for applications that occur at constant pressure. For example, in explosives research Helmholtz free energy is often used, since explosive reactions by their nature induce pressure changes. It is also frequently used to define fundamental equations of state of pure substances.

The concept of free energy was developed by Hermann von Helmholtz, a German physicist, and first presented in 1882 in a lecture called "On the thermodynamics of chemical processes". From the German word Arbeit (work), the International Union of Pure and Applied Chemistry (IUPAC) recommends the symbol A and the name Helmholtz energy. In physics, the symbol F is also used in reference to free energy or Helmholtz function.

Definition

The Helmholtz free energy is defined as

$${\displaystyle F\equiv U-TS,}$$

where

• F is the Helmholtz free energy (sometimes also called A, particularly in the field of chemistry) (SI: joules, CGS: ergs),
• U is the internal energy of the system (SI: joules, CGS: ergs),
• T is the absolute temperature (kelvins) of the surroundings, modelled as a heat bath,
• S is the entropy of the system (SI: joules per kelvin, CGS: ergs per kelvin).

The Helmholtz energy is the Legendre transformation of the internal energy U, in which temperature replaces entropy as the independent variable.

Formal development

The first law of thermodynamics in a closed system provides

${\displaystyle \mathrm{d}U=\delta Q+\delta W,}$

where ${\displaystyle U}$ is the internal energy, ${\displaystyle \delta Q}$ is the energy added as heat, and ${\displaystyle \delta W}$ is the work done on the system. The second law of thermodynamics for a reversible process yields ${\displaystyle \delta Q=T\mathrm{d}S}$. In case of a reversible change, the work done can be expressed as ${\displaystyle \delta W=-p\mathrm{d}V}$ (ignoring electrical and other non-PV work) and so:

${\displaystyle \mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V.}$

Applying the product rule for differentiation to ${\displaystyle \mathrm{d}(TS)=T\mathrm{d}S+S\mathrm{d}T}$, it follows

${\displaystyle \mathrm{d}U=\mathrm{d}(TS)-S\mathrm{d}T-p\mathrm{d}V,}$

and

${\displaystyle \mathrm{d}(U-TS)=-S\mathrm{d}T-p\mathrm{d}V.}$

The definition of ${\displaystyle F=U-TS}$ enables to rewrite this as

${\displaystyle \mathrm{d}F=-S\mathrm{d}T-p\mathrm{d}V.}$

Because F is a thermodynamic function of state, this relation is also valid for a process (without electrical work or composition change) that is not reversible.

Minimum free energy and maximum work principles

The laws of thermodynamics are only directly applicable to systems in thermal equilibrium. If we wish to describe phenomena like chemical reactions, then the best we can do is to consider suitably chosen initial and final states in which the system is in (metastable) thermal equilibrium. If the system is kept at fixed volume and is in contact with a heat bath at some constant temperature, then we can reason as follows.

Since the thermodynamical variables of the system are well defined in the initial state and the final state, the internal energy increase ${\displaystyle \mathrm{\Delta }U}$, the entropy increase ${\displaystyle \mathrm{\Delta }S}$, and the total amount of work that can be extracted, performed by the system, ${\displaystyle W}$, are well defined quantities. Conservation of energy implies

${\displaystyle \mathrm{\Delta }{U}_{\text{bath}}+\mathrm{\Delta }U+W=0.}$

The volume of the system is kept constant. This means that the volume of the heat bath does not change either, and we can conclude that the heat bath does not perform any work. This implies that the amount of heat that flows into the heat bath is given by

${\displaystyle Q_{\text{bath}}=\mathrm{\Delta }{U}_{\text{bath}}=-(\mathrm{\Delta }U+W).}$

The heat bath remains in thermal equilibrium at temperature T no matter what the system does. Therefore, the entropy change of the heat bath is

${\displaystyle \mathrm{\Delta }{S}_{\text{bath}}=\frac{Q_{\text{bath}}}{T}=-\frac{\mathrm{\Delta }U+W}{T}.}$

The total entropy change is thus given by

${\displaystyle \mathrm{\Delta }{S}_{\text{bath}}+\mathrm{\Delta }S=-\frac{\mathrm{\Delta }U-T\mathrm{\Delta }S+W}{T}.}$

Since the system is in thermal equilibrium with the heat bath in the initial and the final states, T is also the temperature of the system in these states. The fact that the system's temperature does not change allows us to express the numerator as the free energy change of the system:

${\displaystyle \mathrm{\Delta }{S}_{\text{bath}}+\mathrm{\Delta }S=-\frac{\mathrm{\Delta }F+W}{T}.}$

Since the total change in entropy must always be larger or equal to zero, we obtain the inequality

${\displaystyle W\le -\mathrm{\Delta }F.}$

We see that the total amount of work that can be extracted in an isothermal process is limited by the free-energy decrease, and that increasing the free energy in a reversible process requires work to be done on the system. If no work is extracted from the system, then

${\displaystyle \mathrm{\Delta }F\le 0,}$

and thus for a system kept at constant temperature and volume and not capable of performing electrical or other non-PV work, the total free energy during a spontaneous change can only decrease.

This result seems to contradict the equation dF = −S dT − P dV, as keeping T and V constant seems to imply dF = 0, and hence F = constant. In reality there is no contradiction: In a simple one-component system, to which the validity of the equation dF = −S dT − P dV is restricted, no process can occur at constant T and V, since there is a unique P(T, V) relation, and thus T, V, and P are all fixed. To allow for spontaneous processes at constant T and V, one needs to enlarge the thermodynamical state space of the system. In case of a chemical reaction, one must allow for changes in the numbers N_j of particles of each type j. The differential of the free energy then generalizes to

${\displaystyle dF=-SdT-PdV+\sum _j{\mu }_{j}d{N}_j,}$

where the ${\displaystyle N_j}$ are the numbers of particles of type j and the ${\displaystyle {\mu }_j}$ are the corresponding chemical potentials. This equation is then again valid for both reversible and non-reversible changes. In case of a spontaneous change at constant T and V, the last term will thus be negative.

In case there are other external parameters, the above relation further generalizes to

${\displaystyle dF=-SdT-\sum _i{X}_{i}d{x}_i+\sum _j{\mu }_{j}d{N}_j.}$

Here the ${\displaystyle x_i}$ are the external variables, and the ${\displaystyle X_i}$ the corresponding generalized forces.

Relation to the canonical partition function

A system kept at constant volume, temperature, and particle number is described by the canonical ensemble. The probability of finding the system in some energy eigenstate r, for any microstate i, is given by

$${\displaystyle P_r=\frac{e^{-\beta E_r}}{Z},}$$

where

• ${\displaystyle \beta =\frac{1}{kT},}$
• ${\displaystyle E_r}$ is the energy of accessible state ${\displaystyle r}$
• $Z=\sum _i{e}^{-\beta E_i}.$

Z is called the partition function of the system. The fact that the system does not have a unique energy means that the various thermodynamical quantities must be defined as expectation values. In the thermodynamical limit of infinite system size, the relative fluctuations in these averages will go to zero.

The average internal energy of the system is the expectation value of the energy and can be expressed in terms of Z as follows:

${\displaystyle U\equiv ⟨E⟩=\sum _r{P}_r{E}_r=\sum _r\frac{e^{-\beta E_r}{E}_r}{Z}=\sum _r\frac{-\frac{\mathrm{\partial }}{\mathrm{\partial }\beta }{e}^{-\beta E_r}}{Z}=\frac{-\frac{\mathrm{\partial }}{\mathrm{\partial }\beta }\sum _r{e}^{-\beta E_r}}{Z}=-\frac{\mathrm{\partial }\log Z}{\mathrm{\partial }\beta }.}$

If the system is in state r, then the generalized force corresponding to an external variable x is given by

${\displaystyle X_r=-\frac{\mathrm{\partial }{E}_r}{\mathrm{\partial }x}.}$

The thermal average of this can be written as

${\displaystyle X=\sum _r{P}_r{X}_r=\frac{1}{\beta }\frac{\mathrm{\partial }\log Z}{\mathrm{\partial }x}.}$

Suppose that the system has one external variable ${\displaystyle x}$. Then changing the system's temperature parameter by ${\displaystyle d\beta }$ and the external variable by ${\displaystyle dx}$ will lead to a change in ${\displaystyle \log Z}$:

${\displaystyle d(\log Z)=\frac{\mathrm{\partial }\log Z}{\mathrm{\partial }\beta }d\beta +\frac{\mathrm{\partial }\log Z}{\mathrm{\partial }x}dx=-Ud\beta +\beta Xdx.}$

If we write ${\displaystyle Ud\beta }$ as

${\displaystyle Ud\beta =d(\beta U)-\beta dU,}$

we get

${\displaystyle d(\log Z)=-d(\beta U)+\beta dU+\beta Xdx.}$

This means that the change in the internal energy is given by

${\displaystyle dU=\frac{1}{\beta }d(\log Z+\beta U)-Xdx.}$

In the thermodynamic limit, the fundamental thermodynamic relation should hold:

${\displaystyle dU=TdS-Xdx.}$

This then implies that the entropy of the system is given by

${\displaystyle S=k\log Z+\frac{U}{T}+c,}$

where c is some constant. The value of c can be determined by considering the limit T → 0. In this limit the entropy becomes ${\displaystyle S=k\log {\mathrm{\Omega }}_0}$, where ${\displaystyle {\mathrm{\Omega }}_0}$ is the ground-state degeneracy. The partition function in this limit is ${\displaystyle {\mathrm{\Omega }}_0{e}^{-\beta U_0}}$, where ${\displaystyle U_0}$ is the ground-state energy. Thus, we see that ${\displaystyle c=0}$ and that

${\displaystyle F=-kT\log Z.}$

Relating free energy to other variables

Combining the definition of Helmholtz free energy

${\displaystyle F=U-TS}$

along with the fundamental thermodynamic relation

$${\displaystyle dF=-SdT-PdV+\mu dN,}$$

one can find expressions for entropy, pressure and chemical potential:

${\displaystyle S={-\left(\frac{\mathrm{\partial }F}{\mathrm{\partial }T}\right)|}_{V,N},P={-\left(\frac{\mathrm{\partial }F}{\mathrm{\partial }V}\right)|}_{T,N},\mu ={\left(\frac{\mathrm{\partial }F}{\mathrm{\partial }N}\right)|}_{T,V}.}$

These three equations, along with the free energy in terms of the partition function,

${\displaystyle F=-kT\log Z,}$

allow an efficient way of calculating thermodynamic variables of interest given the partition function and are often used in density of state calculations. One can also do Legendre transformations for different systems. For example, for a system with a magnetic field or potential, it is true that

${\displaystyle m={-\left(\frac{\mathrm{\partial }F}{\mathrm{\partial }B}\right)|}_{T,N},V={\left(\frac{\mathrm{\partial }F}{\mathrm{\partial }Q}\right)|}_{N,T}.}$

Bogoliubov inequality

Computing the free energy is an intractable problem for all but the simplest models in statistical physics. A powerful approximation method is mean-field theory, which is a variational method based on the Bogoliubov inequality. This inequality can be formulated as follows.

Suppose we replace the real Hamiltonian ${\displaystyle H}$ of the model by a trial Hamiltonian ${\displaystyle \tilde{H}}$, which has different interactions and may depend on extra parameters that are not present in the original model. If we choose this trial Hamiltonian such that

${\displaystyle ⟨\tilde{H}⟩=⟨H⟩,}$

where both averages are taken with respect to the canonical distribution defined by the trial Hamiltonian ${\displaystyle \tilde{H}}$, then the Bogoliubov inequality states

${\displaystyle F\le \tilde{F},}$

where ${\displaystyle F}$ is the free energy of the original Hamiltonian, and ${\displaystyle \tilde{F}}$ is the free energy of the trial Hamiltonian. We will prove this below.

By including a large number of parameters in the trial Hamiltonian and minimizing the free energy, we can expect to get a close approximation to the exact free energy.

The Bogoliubov inequality is often applied in the following way. If we write the Hamiltonian as

${\displaystyle H=H_0+\mathrm{\Delta }H,}$

where ${\displaystyle H_0}$ is some exactly solvable Hamiltonian, then we can apply the above inequality by defining

${\displaystyle \tilde{H}=H_0+⟨\mathrm{\Delta }H{⟩}_0.}$

Here we have defined ${\displaystyle ⟨X{⟩}_0}$ to be the average of X over the canonical ensemble defined by ${\displaystyle H_0}$. Since ${\displaystyle \tilde{H}}$ defined this way differs from ${\displaystyle H_0}$ by a constant, we have in general

${\displaystyle ⟨X{⟩}_0=⟨X⟩.}$

where ${\displaystyle ⟨X⟩}$ is still the average over ${\displaystyle \tilde{H}}$, as specified above. Therefore,

${\displaystyle ⟨\tilde{H}⟩=⟨H_0+⟨\mathrm{\Delta }H⟩⟩=⟨H⟩,}$

and thus the inequality

${\displaystyle F\le \tilde{F}}$

holds. The free energy ${\displaystyle \tilde{F}}$ is the free energy of the model defined by ${\displaystyle H_0}$ plus ${\displaystyle ⟨\mathrm{\Delta }H⟩}$. This means that

${\displaystyle \tilde{F}=⟨H_0{⟩}_0-T{S}_0+⟨\mathrm{\Delta }H{⟩}_0=⟨H{⟩}_0-T{S}_0,}$

and thus

${\displaystyle F\le ⟨H{⟩}_0-T{S}_0.}$

Proof of the Bogoliubov inequality

For a classical model we can prove the Bogoliubov inequality as follows. We denote the canonical probability distributions for the Hamiltonian and the trial Hamiltonian by ${\displaystyle P_r}$ and ${\displaystyle {\tilde{P}}_r}$, respectively. From Gibbs' inequality we know that:

${\displaystyle \sum _r{\tilde{P}}_r\log \left({\tilde{P}}_r\right)\ge \sum _r{\tilde{P}}_r\log \left(P_r\right)}$

holds. To see this, consider the difference between the left hand side and the right hand side. We can write this as:

${\displaystyle \sum _r{\tilde{P}}_r\log \left(\frac{{\tilde{P}}_r}{P_r}\right)}$

Since

${\displaystyle \log \left(x\right)\ge 1-\frac{1}{x}}$

it follows that:

${\displaystyle \sum _r{\tilde{P}}_r\log \left(\frac{{\tilde{P}}_r}{P_r}\right)\ge \sum _r\left({\tilde{P}}_r-P_r\right)=0}$

where in the last step we have used that both probability distributions are normalized to 1.

We can write the inequality as:

${\displaystyle ⟨\log \left({\tilde{P}}_r\right)⟩\ge ⟨\log \left(P_r\right)⟩}$

where the averages are taken with respect to ${\displaystyle {\tilde{P}}_r}$. If we now substitute in here the expressions for the probability distributions:

${\displaystyle P_r=\frac{\exp \left[-\beta H\left(r\right)\right]}{Z}}$

and

${\displaystyle {\tilde{P}}_r=\frac{\exp \left[-\beta \tilde{H}\left(r\right)\right]}{\tilde{Z}}}$

we get:

${\displaystyle ⟨-\beta \tilde{H}-\log \left(\tilde{Z}\right)⟩\ge ⟨-\beta H-\log \left(Z\right)⟩}$

Since the averages of ${\displaystyle H}$ and ${\displaystyle \tilde{H}}$ are, by assumption, identical we have:

${\displaystyle F\le \tilde{F}}$

Here we have used that the partition functions are constants with respect to taking averages and that the free energy is proportional to minus the logarithm of the partition function.

We can easily generalize this proof to the case of quantum mechanical models. We denote the eigenstates of ${\displaystyle \tilde{H}}$ by ${\displaystyle |r⟩}$. We denote the diagonal components of the density matrices for the canonical distributions for ${\displaystyle H}$ and ${\displaystyle \tilde{H}}$ in this basis as:

${\displaystyle P_r=⟨r\left|\frac{\exp \left[-\beta H\right]}{Z}\right|r⟩}$

and

${\displaystyle {\tilde{P}}_r=⟨r\left|\frac{\exp \left[-\beta \tilde{H}\right]}{\tilde{Z}}\right|r⟩=\frac{\exp \left(-\beta {\tilde{E}}_r\right)}{\tilde{Z}}}$

where the ${\displaystyle {\tilde{E}}_r}$ are the eigenvalues of ${\displaystyle \tilde{H}}$

We assume again that the averages of H and ${\displaystyle \tilde{H}}$ in the canonical ensemble defined by ${\displaystyle \tilde{H}}$ are the same:

${\displaystyle ⟨\tilde{H}⟩=⟨H⟩}$

where

${\displaystyle ⟨H⟩=\sum _r{\tilde{P}}_r⟨r\left|H\right|r⟩}$

The inequality

${\displaystyle \sum _r{\tilde{P}}_r\log \left({\tilde{P}}_r\right)\ge \sum _r{\tilde{P}}_r\log \left(P_r\right)}$

still holds as both the ${\displaystyle P_r}$ and the ${\displaystyle {\tilde{P}}_r}$ sum to 1. On the l.h.s. we can replace:

${\displaystyle \log \left({\tilde{P}}_r\right)=-\beta {\tilde{E}}_r-\log \left(\tilde{Z}\right)}$

On the right-hand side we can use the inequality

${\displaystyle {⟨\exp \left(X\right)⟩}_r\ge \exp \left({⟨X⟩}_r\right)}$

where we have introduced the notation

${\displaystyle {⟨Y⟩}_r\equiv ⟨r\left|Y\right|r⟩}$

for the expectation value of the operator Y in the state r. See here for a proof. Taking the logarithm of this inequality gives:

${\displaystyle \log \left[{⟨\exp \left(X\right)⟩}_r\right]\ge {⟨X⟩}_r}$

This allows us to write:

${\displaystyle \log \left(P_r\right)=\log \left[{⟨\exp \left(-\beta H-\log \left(Z\right)\right)⟩}_r\right]\ge {⟨-\beta H-\log \left(Z\right)⟩}_r}$

The fact that the averages of H and ${\displaystyle \tilde{H}}$ are the same then leads to the same conclusion as in the classical case:

${\displaystyle F\le \tilde{F}}$

Generalized Helmholtz energy

In the more general case, the mechanical term ${\displaystyle p\mathrm{d}V}$ must be replaced by the product of volume, stress, and an infinitesimal strain:

${\displaystyle \mathrm{d}F=V\sum _{ij}{\sigma }_{ij}\mathrm{d}{\epsilon }_{ij}-S\mathrm{d}T+\sum _i{\mu }_i\mathrm{d}{N}_i,}$

where ${\displaystyle {\sigma }_{ij}}$ is the stress tensor, and ${\displaystyle {\epsilon }_{ij}}$ is the strain tensor. In the case of linear elastic materials that obey Hooke's law, the stress is related to the strain by

${\displaystyle {\sigma }_{ij}=C_{ijkl}{\epsilon }_{kl},}$

where we are now using Einstein notation for the tensors, in which repeated indices in a product are summed. We may integrate the expression for ${\displaystyle \mathrm{d}F}$ to obtain the Helmholtz energy:

${\displaystyle \begin{array}{rl}F& =\frac{1}{2}V{C}_{ijkl}{\epsilon }_{ij}{\epsilon }_{kl}-ST+\sum _i{\mu }_i{N}_i\\ & =\frac{1}{2}V{\sigma }_{ij}{\epsilon }_{ij}-ST+\sum _i{\mu }_i{N}_i.\end{array}}$

Application to fundamental equations of state

The Helmholtz free energy function for a pure substance (together with its partial derivatives) can be used to determine all other thermodynamic properties for the substance. See, for example, the equations of state for water, as given by the IAPWS in their IAPWS-95 release.

Application to training auto-encoders

Hinton and Zemel "derive an objective function for training auto-encoder based on the minimum description length (MDL) principle". "The description length of an input vector using a particular code is the sum of the code cost and reconstruction cost. They define this to be the energy of the code. Given an input vector, they define the energy of a code to be the sum of the code cost and the reconstruction cost." The true expected combined cost is

${\displaystyle F=\sum _i{p}_i{E}_i-H,}$

"which has exactly the form of Helmholtz free energy".

Biophilic design

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Biophilic_design 

 

Biophilic learning space at Ohalo College in Israel.

Biophilic design is a concept used within the building industry to increase occupant connectivity to the natural environment through the use of direct nature, indirect nature, and space and place conditions. Used at both the building and city-scale, it is argued that this idea has health, environmental, and economic benefits for building occupants and urban environments, with few drawbacks. Although its name was coined in recent history, indicators of biophilic design have been seen in architecture from as far back as the Hanging Gardens of Babylon.

Biophilia hypothesis

Main article: Biophilia hypothesis

The word “Biophilia” was first introduced by a psychoanalyst named Erich Fromm who stated that biophilia is the “passionate love of life and of all that is alive…whether in a person, a plant, an idea, or a social group” in his book The Anatomy of Human Destructiveness in 1973. Fromm's approach was that of a psychoanalyst (a person who studies the unconscious mind) and presented a broad spectrum as he called biophilia a biologically normal instinct.

The term has been used since by many scientists, and philosophers overall being adapted to several different areas of study. Some notable mentions of biophilia include Edward O. Wilson's book Biophilia (1984) where he took a biologist's approach and first coined the “Biophilia hypothesis” and popularized the notion. Wilson defined biophilia as “the innate tendency to focus on life and lifelike processes”, claiming a link with nature is not only physiological (as Fromm suggested) but has a genetic basis. The biophilia hypothesis is the idea that humans have an inherited need to connect to nature and other biotic forms due to our evolutionary dependence on it for survival and personal fulfillment. This idea is relevant in daily life – humans travel and spend money to sightsee in national parks and nature preserves, relax on beaches, hike mountains, and explore jungles. Further, many sports revolve around nature such as skiing, mountain biking, and surfing. From a home perspective, people are more likely to spend more on houses that have views of nature; buyers are willing to spend 7% more on homes with excellent landscaping, 58% more on properties that look at water, and 127% more on those that are waterfront. Humans also value companionship with animals. In America 60.2 million people own dogs and 47.1 million own cats.

Biophobia

While biophilia refers to the inherent need to experience and love nature, biophobia is human's inherited fear of nature and animals. In the case of modern life, humans urge to separate ourselves from nature and move towards technology; a cultural drive where people tend to associate with human artifacts, interests, and managed activities. Some anxieties of the natural environment are inherited from threats seen in anthropocentric evolution: this includes fear of snakes, spiders, and blood. In relation to buildings, biophobia can be induced through the use of bright colors, heights, enclosed spaces, darkness, and large open spaces are major contributors to occupant discomfort.

Dimensions

Considered as one of the pioneers of biophilic design, Stephen Kellert has created a framework where nature in the built environment is used in a way that satisfies human needs – his principles are meant to celebrate and show respect for nature, and provide an enriching urban environment that is multisensory. The dimensions and attributes that define Kellert's biophilic framework are below.

Direct experience of nature

An example of a green wall at Simon Fraser University, in British Columbia

Direct experience refers to tangible contact with natural features:

• Light: Allows orientation of time of day and season, and is attributed to wayfinding and comfort; light can also cause natural patterns and form, movements and shadows. In design, this can be applied through clerestories, reflective materials, skylights, glass, and atriums. This provides well-being and interest from occupants.
• Air: Ventilation, temperature, and humidity are felt through air. Such conditions can be applied through the use of windows and other passive strategies, but most importantly the variation in these elements can promote occupant comfort and productivity.
• Water: Water is multisensory and can be used in buildings to provide movement, sounds, touch, and sight. In design it can be incorporated through water bodies, fountains, wetlands, and aquariums; people have a strong connection to water and when used, it can decrease stress and increase health, performance, and overall satisfaction.
• Plants: Bringing vegetation to the exterior and interior spaces of the building provides a direct relationship to nature. This should be abundant (i.e., make use of green walls or many potted plants) and some vegetation should flower; plants have been proven to increase physical health, performance, and productivity and reduce stress.
• Animals: While hard to achieve, it can be done through aquariums, gardens, animal feeders, and green roofs. This interaction with promotes interest, mental stimulation, and pleasure.
• Weather: Weather can be observed directly through windows and transitional spaces, but it can also be simulated through the manipulation of air within the space; awareness of weather signified human fitness and survival in ancient times and now promotes awareness and mental stimulation.
• Natural landscapes: This is done through creating self-sustaining ecosystems into the built environment. Given human evolution and history, people tend to enjoy savannah-like landscapes as they depict spaciousness and an abundance of natural life. Contact with these types of environments can be done through vistas and or direct interactions such as gardens. Such landscapes are known to increase occupant satisfaction.
• Fire: This natural element is hard to incorporate, however when implemented correctly into the building, it provides color, warmth, and movement, all of which are appealing and pleasing to occupants.

Indirect experience of nature

Indirect experience refers to contact with images and or representations of nature:

• Images of Nature: This has been proven to be emotionally and intellectually satisfying to occupants; images of nature can be implemented through paintings, photos, sculptures, murals, videos, etcetera.
• Natural Materials: People prefer natural materials as they can be mentally stimulating. Natural materials are susceptible to the patina of time; this change invokes responses from people. These materials can be incorporated into buildings through the use of wood and stone. Interior design can use natural fabrics and furnishings. Leather has often been included as recommended Biophilic material however with the awareness of animal agriculture (leather being a co-product of the meat industry) as a major contributor to climate change faux, or plant-based, leathers created from mushroom, pineapple skin, or cactus are now seen as viable alternatives. It is also seen that to feel, and be, closer to nature and animals to destroy them in the pursuit of this is counter-productive and in conflict with the philosophy of Biophilia.
• Natural Colors: Natural colors or “earth-tones”, are those that are commonly found in nature and are often subdued tones of brown, green, and blue. When using colors in buildings, they should represent these natural tones. Brighter colors should only be used sparingly – one study found that red flowers on plants were found to be fatiguing and distracting by occupants.
• Simulations of Natural Light and Air: In areas where natural forms of ventilation and light cannot be achieved, creative use of interior lighting and mechanical ventilation can be used to mimic these natural features. Designers can do this through variations in lighting through different lighting types, reflective mediums, and natural geometries that the fixture can shine through; natural airflow can be imitated through mild changes in temperature, humidity, and air velocity.
• Naturalistic Shapes: Natural shapes and forms can be achieved in architectural design through columns and nature-based patterns on facades - including these different elements into spaces can change a static space into an intriguing and appealing complex area.
• Evoking Nature: This uses characteristics found in nature to influence the structural design of the project. These may be things that may not occur in nature, rather elements that represent natural landscapes such as mimicking different plant heights found in ecosystems, and or mimicking particular animal, water, or plant features.
• Information Richness: This can be achieved by providing complex, yet not noisy environments that invoke occupant curiosity and thought. Many ecosystems are complex and filled with different abiotic and biotic elements – in such the goal of this attribute is to include these elements into the environment of the building.
• Change and the Patina of Time: People are intrigued by nature and how it changes, adapts, and ages over time, much like ourselves. In buildings, this can be accomplished by using organic materials that are susceptible to weathering and color change – this allows for us to observe slight changes in our built environment over time.
  

• Example of a natural fractal
• Natural Geometries: The design of facades or structural components can include the use of repetitive, varied patterns that are seen in nature (fractals). These geometries can also have hierarchically organized scales and winding flow rather than be straight with harsh angles. For instance, commonly used natural geometries are the honeycomb pattern and ripples found in water.
• Biomimicry: This is a design strategy that imitates uses found in nature as solutions for human and technical problems. Using these natural functions in construction can entice human creativity and consideration of nature.

Experience of space and place

The experience of space and place uses spatial relationships to enhance well-being:

Thorncrown Chapel is often seen as a model of biophilic design due to it having all three of Kellert's experiences.

• Prospect and Refuge: Refuge refers to the building's ability to provide comfortable and nurturing interiors (alcoves, dimmer lighting), while prospect emphasizes horizons, movement, and sources of danger. Examples of design elements include balconies, alcoves, lighting changes, and areas spaciousness (savannah environment).
• Organized Complexity: This principle is meant to simulate the need for controlled variability; this is done in design through repetition, change, and detail of the building's architecture.
• Integration of Parts: When different parts comprise a whole, it provides satisfaction for occupants: design elements include interior spaces using clear boundaries and or the integration of a central focal point.
• Transitional Spaces: This element aims to connect interior spaces with the outside or create comfort by providing access from one space to another environment through the use of porches, decks, atriums, doors, bridges, fenestrations, and foyers.
• Mobility: The ability for people to comfortably move between spaces, even when complex; it provides the feeling of security for occupants and can be done through making clear points of entry and egress.
• Cultural and Ecological Attachment to Place: Creating a cultural sense of place in the built environment creates human connection and identity. This is done by incorporating the area's geography and history into the design. Ecological identity is done through the creation of ecosystems that promote the use of native flora and fauna.

Each of these experiences are meant to be considered individually when using biophilia in projects, as there is no one right answer for one building type. Each building's architect(s) and project owner(s) must collaborate to include the biophilic principles they believe fit within their scope and most effectively reach their occupants.

City-scale

Timothy Beatley believes the key objective of biophilic cities is to create an environment where the residents want to actively participate in, preserve, and connect with the natural landscape that surrounds them. He established ways to achieve this through a framework of infrastructure, governance, knowledge, and behavior; these dimensions can also be indicators of existing biophilic attributes that already exist in current cities.

• Biophilic Conditions and Infrastructure: The idea that a certain number of people at any given time should be near a green space or park. This can be done through the creation of integrated ecological networks and walking trails throughout the city, the designation of certain portions of land area for vegetation and forests, green and biophilic building design features, and the use of flora and fauna throughout the city.
• Biophilic Activities: This refers to the increased amount of time spent outside and visiting parks, longer outdoor periods at schools, improved foot traffic across the city, improved participation in community gardens and conservatory clubs, larger participation in local volunteer efforts.
• Biophilic Attitudes and Knowledge: In areas with urban biophilic design elements, there will be an improved number of residents who care about nature and can identify local native species; resident curiosity of their local ecosystems also increases.
• Biophilic Institutions and Governance: Local government bodies allocate part of the budget to nature and biophilic activities. Indicators of this include increased regulation that requires more green and biophilic design principles, grant programs that promote the use of nature and biophilia, the inclusion of natural history museums and educational programs, and increased number of nature non-governmental organizations and community groups.

Benefits

Biophilic design is argued to have a wealth of benefits for building occupants and urban environments through improving connections to nature. For cities, many believe the biggest proponent of the concept is its ability to make the city more resilient to any environmental stressor it may face.

Health benefits

Catherine Ryan, et al. found that elements such as nature sounds, improved mental health 37% faster than traditional urban noise after stressor exposure; the same study found that when surgery patients were exposed to aromatherapy, 45% used less morphine and 56% used fewer painkillers overall. Another study by Kaitlyn Gillis and Birgitta Gatersleben found that the inclusion of plants in interior environments reduce stress and increase pain tolerance; the use of water elements and incorporating views of nature are also mentally restorative for occupants. When researching the effects of biophilia on hospital patients, Peter Newman and Jana Soderlund found that by increasing vista quality in hospital rooms depression and pain in patients is reduced, which in turn shortened hospital stays from 3.67 days to 2.6 days. In biophilic cities, Andrew Dannenberg, et al. indicated that there are higher levels of social connectivity and better capability to handle life crises; this has resulted in lower crime rate levels of violence and aggression. The same study found that implementing outdoor facilities such as impromptu gymnasiums like the “Green Gym” in the United Kingdom, allow people to help clear overgrown vegetation, build walking paths, plant foliage, and more readily exercise (walking, running, climbing, etc.); this has been proven to build social capital, increase physical activity, better mental health and quality of life. Further, Dannenberg, et al. also found that children growing up in green neighborhoods are seen to have lower levels of asthma; decreased mortality rates and health disparities between the wealthy and poor were also observed in greener neighborhoods.

Environmental benefits

Example of a rain garden that increases infiltration

Some argue that by adding physical natural elements, such as plants, trees, rain gardens, and green roofs, to the built environment, buildings and cities can manage stormwater runoff better as there are fewer impervious surfaces and better infiltration. To maintain these natural systems in a cost-effective way, excess greywater can be reused to water the plants and greenery; vegetative walls and roofs also decrease polluted water as the plants act as biofilters. Adding greenery also reduces carbon emissions, the heat island effect, and increases biodiversity. Carbon is reduced through carbon sequestration in the plant's roots during photosynthesis. Green and high albedo rooftops and facades, and shading of streets and structures using vegetation can reduce the amount of heat absorption normally found in asphalt or dark surfaces – this can reduce heating and cooling needs by 25% and reduce temperature fluctuations by 50%. Further, adding green facades can increase the biodiversity of an area if native species are planted - the Khoo Teck Puat Hospital in Singapore has seen a resurgence of 103 species of butterflies onsite, thanks to their use of vegetation throughout the exterior of the building.

Economic benefits

Biophilia may have slightly higher costs due to the addition of natural elements that require maintenance, higher-priced organic items, etc., however, the perceived health and environmental benefits are believed to negate this. Peter Newman found that by adding biophilic design and landscapes, cities like New York City can see savings nearing $470 million due to increased worker productivity and $1.7 billion from reduced crime expenses. They also found that storefronts on heavily vegetated streets increased foot traffic and attracted consumers that were likely to spend 25% more; the same study showed that increasing daylighting through skylights in a store increase sales by 40% +/- 7%. Properties with biophilic design also benefit from higher selling prices, with many selling at 16% more than conventional buildings.

Sustainability and resilience

Beatley's biophilic pathways for urban resilience

On the urban scale, Timothy Beatley believes that biophilic design will allow cities to better adapt to stresses that occur from changes in climate and thus, local environments. To better show this, he created a biophilic cities framework, where pathways can be taken to increase the resilience and sustainability of cities. This includes three sections: Biophilic Urbanism - the physical biophilic and green measures that can be taken to increase the resilience of the city, Adaptive Capacity - how the community's behaviors will adapt as a result of these physical changes, and Resilient Outcomes - what can happen if both of these steps are achieved.

Under the Biophilic Urbanism section, one of the ways a city can increase resilience is by pursuing the biophysical pathway – by safeguarding and promoting the inclusion of natural systems, the natural protective barrier of the city is increased. For example, New Orleans is a city that has built over its natural wet plains and has exposed themselves to flooding. It is estimated that if they kept the bayous intact, the city could save $23 billion yearly in storm protection.

In the Adaptive Capacity section, Beatley states that the commitment to place and home pathway creates stimulating and interesting nature environments for residents – this will create stronger bonds to home, which will increase the likelihood that citizens will take care of where they live. He goes further in saying that in times of shock or stress, these people are more likely to rebuild and or support the community instead of fleeing. This may also increase governmental action to protect the city from future disasters.

By achieving Biophilic Urbanism and Adaptive Capacity, Beatley believes that one of the biggest resilient outcomes of this framework will be increased adaptability of the residents. Because the steps leading to resilience encourage people to be outside walking and participating in activities, the citizens become healthier and more physically fit; it has been found that those who take walks in nature experience decreased depression, anger, and increased vigor, versus those who walk in interior environments.

Use in building standards

Given the increased information supporting the benefits of biophilic design, organizations are beginning to incorporate the concept into their standards and rating systems to encourage building professionals to use biophilia in their projects. As of now, the most prominent supporters of biophilic design are the WELL Building Standard and the Living Building Challenge.

WELL Building Standard

The International WELL Building Institute uses biophilic design in their WELL Standard as a qualitative and quantitive metric. The qualitative metric must incorporate nature (environmental elements, natural lighting, and spatial qualities), natural patterns, and nature interaction within and outside the building; these efforts must be documented through professional narrative to be considered for certification. For the quantitative portion, projects must have outdoor biophilia (25% of the project must have accessible landscaped grounds and or rooftop gardens and 70% of that 25% must have plantings), indoor biophilia (plant beds and pots must cover 1% of the floor area and plant walls must cover 2% of the floor area), and water features (projects over 100,000 sqft must have a water feature that is either 1.8 m in height or 4 m² in floor area). Verification is enforced through assurance letters by the architects and owners, and by on-site spot checks. Generally, both metric types can be applied to every building type the WELL Standard addresses, with two exceptions: core and shell construction does not need to include quantitative interior biophilia and existing interiors do not need to include qualitative nature interaction.

Living Building Challenge

Main article: Living Building Challenge

The International Living Future Institute is the creator of the living building challenge – a rigorous building standard that aims to maximize building performance. This standard classifies the use of a biophilic environment as an imperative element in their health and happiness section. The living building challenge requires that a framework be created that shows the following: how the project will incorporate nature through environmental features, light, and space, natural shapes and forms, natural patterns, and place-based relationships. The challenge also requires that the occupants be able to connect to nature directly through interaction within the interior and exterior of the building. These are then verified through a preliminary audit procedure.

Criticisms

Biophilic design is considered young, as it has not been implemented in modern building projects for a long period of time. Because of this, there has been little research that explores the long-term challenges, negatives, and even benefits of biophilia in buildings and cities. Other concerns are the initial and maintenance costs of projects that implement expensive biophilic design principles. This could be due to the lack of research that was discussed above, as there is little information regarding payback periods for investors. Another issue could be the prices for the technology needed, however this should eventually lower as the concept becomes more commonplace.

Building-scale examples of application

Church of Mary Magdalene

Exterior view of Maria Magdalena Church Jerusalem

 

Main article: Church of Mary Magdalene

The Church of Mary Magdalene is in Jerusalem and was consecrated in 1888. This church's architecture is biophilic in that it contains natural geometries, organized complexity, information richness, and organic forms (onion-shaped domes) and materials. On the exterior, complexity and order are shown through the repetitive use of domes, their scale, and placement. Inside, the church experiences symmetry and a savannah-like environment through its vaulting and domes – the columns also have leaf-like fronds, which represents images of nature. Prospect is explored through raised ceilings that have balconies and increased lighting; refuge is experienced in lower areas, where there are reduced lighting and alcoves and throughout, where small windows are encased by thick walls.

Fallingwater

Main article: Fallingwater

 

Frank Lloyd Wright's Fallingwater

Fallingwater, one of Frank Lloyd Wright's most famous buildings, exemplifies many biophilic features. The home has human-nature connectivity through the integrative use of the waterfall and stream in its architecture - the sound from these water features can be heard throughout the inside of the home. This allows visitors to feel like they are “participating” in nature rather than “spectating” it like they would be if the waterfall were downstream. In addition, the structure is built around existing foliage and encompasses the local geology by incorporating a large rock in the center of the living room. There are also many glass walls to connect the occupants to the surrounding woods and nature that is outdoors. To better the flow of the space, Wright included many transitional spaces in the home (porches and decks); he also enhanced the direct and indirect experiences of nature by using multiple fireplaces and a wealth of organic shapes, colors, and materials. His use of Kellert's biophilic design principles are prominent throughout the structure, even though this home was constructed before these ideas were developed.

Khoo Teck Puat Hospital

Main article: Khoo Teck Puat Hospital

Referred to as a “garden hospital”, KTP has an abundance of native plants and water features that surround its exterior. This inclusion of vegetation has increased the biodiversity of the local ecosystem, bringing butterflies and bird species; the rooftop of the hospital is also used by local residents to grow produce. Unlike many other hospitals, 15% of visitors come to Khoo Teck Puat for recreational reasons such as gardening or relaxing. The design behind this hospital was to increase the productivity of its doctors, wellbeing of its visitors, and increase the healing speed and pain resilience of its patients. To do this, the designers incorporated greenery from the hospital's courtyard to its upper floors, where patients have balconies that are covered in scented foliage. The hospital is centered on the Yishun pond, and like Frank Lloyd Wright's Fallingwater, the architects made this natural feature part of the hospital by having water stream through its courtyard, creating the illusion that the water was “drawn” from the pond. The hospital also utilizes natural ventilation as much as possible in common areas and corridors by orienting them in the direction the north and southeast prevailing winds; this has reduced energy consumption by 60% and increased airflow by 20-30%. This creates thermally adequate environments for patients and medical staff alike. Using Kellert strategies above, it is apparent that most of the strategies used for Khoo Teck Puat are direct nature experiences. The hospital also uses transitional spaces to make occupants more connected to the outdoors and has organized complexity throughout its overall architectural design. KTP has created a sense of place for occupants and neighbors, as it acts as a communal place for both those who work there and live nearby.

Sandy Hook Elementary School

See also: Newton Public Schools

After the disaster that struck Sandy Hook Elementary in 2012, a new school was built to help heal the community and provide a new sense of security for those occupying the space. Major biophilic design parameters that Svigals + Partners included in this project are animal feeders, wetlands, courtyards, natural shapes and patterns, natural materials, transitional spaces, images of nature, natural colors, and use of natural light. The school has incorporated a victory garden that is meant to act as a way of healing for children after the tragedy. The architects wanted the children to feel as if they are learning in the trees so they set the school back at the edge of the woods and surrounded the space with large windows; there are also metaphoric metal trees in the lobby that have reflective metal leaves that refract light onto colored glass. Using Kellert's biophilic framework, it is prevalent that the school utilizes many different nature experiences. The use of wood planks and stone on the outside of the building help enforce indirect experiences of nature because these are natural materials. Further, the interior environment of the school experiences information richness through the architects’ use of light reflection and color. Naturalistic shapes are brought into the interior environment through the metal trees and leaves. For experiences of space and place, Svigals + Partners bring nature into the classroom and school through the placement of windows that act as transitional spaces. The school also has a variety of breezeways, bridges, and pathways for students as they move from one space to another. Direct experiences of nature are enjoyed through water features, large rain gardens, and courtyards found on the property. The animal feeders also act as a way to bring fauna into the area.

City-scale examples of application

Singapore, Singapore

Bishan-Ang Mo Kio Park, Singapore

Nicknamed a “city in a garden”, Singapore has dedicated many resources to make a system of nature preserves, parks and connectors (ex. Southern Ridges), and tree-lined streets that promote the return of wildlife and reduce the heat island effect that is often seen in dense city centers; local governments agree with Kellert and Beatley that daily doses of nature enhance the wellbeing of its citizens. To manage stormwater, Singaporean governments have implemented the Bishan-Ang Mo Kio Park Project, where the old concrete water drains were excavated for the reconstruction of the Kallang River; this allowed residents in the area to enjoy the physiological and physical health benefits of having a green space with water. The reimagining of the park has increased the biodiversity of the local ecosystem, with dragonflies, butterflies, hornbills, and smooth-coated otters returning to the Singaporean region - the river also acts as a natural stormwater management system by increasing infiltration and movement of excess water.

Supertree Grove, Singapore

To increase the immediate presence of nature in the city, Singapore provides subsidies (up to half the installation cost) for those who include vegetative walls, green roofs, sky parks, etc. in their building designs. The city-state also has an impressive number of biophilic buildings and structures. For example, their Gardens by the Bay Project has an installation called the “Supertree Grove”. This urban nature installation has over 160,000 plants that stem from 200 different species installed in the 16 supertrees; many of these urban ‘’trees’’ have sky walkways, observatories, and or solar panels. Lastly, Singapore has implemented efforts to increase community engagement through the creation of over 1,000 community gardens for resident use.

Oslo, Norway

Nature trail in the Maridalen Protected Landscape, Oslo

Oslo is sandwiched between the Oslo Fjord and wooded areas. Woods serve as an important feature to this municipality. More than two-thirds of the city is protected forests; in recent surveys over 81% of Oslo residents said they have gone to these forests at least once in the last year. These forests are protected, as Oslo adheres to ISO14001 for its forest management – the trees are controlled under “living forest” standards, which means that limited harvesting is acceptable. In addition to its extensive forest system, the city compounds its exposure to nature by bringing the natural environment into the urban setting. Being an already compact city (after all, two-thirds is forest) the city allocates around 20% of its urban land to green spaces; the local government is in the process of creating a network of paths to connect these green areas so that citizens can walk and ride their bikes undisturbed. In addition to the expanding park accessibility, the city has also restored the city's river the Akerselva, which runs through Oslo's center. Because the water feature is near sets of dense housing, the city made the river more appealing and accessible to residents by adding waterfalls and nature trails; altogether the city has 365 kilometers worth of nature trails.

Buildings included in Oslo's Barcode Building Project

To connect the city with its fjords, Oslo's government has started the process of putting its roadways underground in tunnels. This, combined with the construction of aesthetically creative architecture (Barcode Project) on the waterfront and promenade foot trails, is transforming this area into a place where residents can experience enjoyment from the unobstructed views of the fjord. Lastly, Oslo has a ‘’Noise Action Plan’’ to help alleviate urban noise levels – some of these areas (mostly recreational) have noise levels as low as 50 dB.